Worksheet
Explore the properties, types, and theorems related to triangles, including congruence and similarity, to solve geometric problems effectively.
Triangles - Practice Worksheet
Strengthen your foundation with key concepts and basic applications.
This worksheet covers essential long-answer questions to help you build confidence in Triangles from Mathematics for Class X (Mathematics).
Basic comprehension exercises
Strengthen your understanding with fundamental questions about the chapter.
Questions
Explain the concept of similar triangles and how they differ from congruent triangles. Provide examples to illustrate your explanation.
Think about the definitions of similarity and congruence in terms of angles and sides.
Solution
Similar triangles are triangles that have the same shape but not necessarily the same size. This means their corresponding angles are equal, and their corresponding sides are in proportion. Congruent triangles, on the other hand, are identical in both shape and size, meaning all corresponding angles and sides are equal. For example, two equilateral triangles of different sizes are similar but not congruent. In real life, similar triangles can be seen in the shadows of objects, where the shadow and the object form triangles that are similar due to the angle of the sun's rays.
Prove the Basic Proportionality Theorem (Thales Theorem) with a diagram and explain its significance.
Draw a triangle with a line parallel to one side and use the properties of similar triangles.
Solution
The Basic Proportionality Theorem states that if a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio. To prove this, consider triangle ABC with a line DE parallel to BC intersecting AB at D and AC at E. By the property of similar triangles, triangles ADE and ABC are similar. Therefore, AD/DB = AE/EC. This theorem is significant as it forms the basis for many other theorems and problems in geometry, including the concept of similar triangles.
Describe the criteria for similarity of triangles and give an example for each criterion.
Recall the conditions under which two triangles are considered similar.
Solution
There are three main criteria for the similarity of triangles: 1) AAA (Angle-Angle-Angle): If the corresponding angles of two triangles are equal, the triangles are similar. For example, two triangles with angles 60°, 60°, 60° are similar. 2) SSS (Side-Side-Side): If the corresponding sides of two triangles are in proportion, the triangles are similar. For example, triangles with sides in the ratio 2:3:4 and 4:6:8 are similar. 3) SAS (Side-Angle-Side): If one angle of a triangle is equal to one angle of another triangle and the sides including these angles are in proportion, the triangles are similar. For example, two triangles with an angle of 50° and the including sides in the ratio 2:3 are similar.
How can the concept of similar triangles be used to find the height of a tree using its shadow? Explain with a diagram.
Consider the properties of similar triangles formed by the objects and their shadows.
Solution
The height of a tree can be found using similar triangles by comparing the shadow of the tree to the shadow of a known object. For instance, if a 1-meter stick casts a 0.5-meter shadow at the same time a tree casts a 10-meter shadow, the triangles formed by the stick and its shadow and the tree and its shadow are similar. Therefore, the height of the tree (h) can be found using the proportion 1/0.5 = h/10, solving which gives h = 20 meters. This method is practical for measuring heights of objects that are difficult to measure directly.
Explain the Pythagorean Theorem and prove it using the concept of similar triangles.
Draw a right-angled triangle and its altitude to form similar triangles.
Solution
The Pythagorean Theorem states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. To prove this using similar triangles, consider a right-angled triangle ABC with the right angle at A. Draw the altitude AD from A to the hypotenuse BC. This creates two smaller triangles, ABD and ADC, which are similar to ABC. By the properties of similar triangles, the ratios of corresponding sides are equal, leading to the equations AB² = BD * BC and AC² = DC * BC. Adding these equations gives AB² + AC² = BD * BC + DC * BC = BC(BD + DC) = BC², thus proving the theorem.
What is the Angle Bisector Theorem? Prove it and explain its application.
Use the properties of parallel lines and similar triangles to prove the theorem.
Solution
The Angle Bisector Theorem states that the angle bisector of a triangle divides the opposite side into two segments that are proportional to the adjacent sides. To prove it, consider triangle ABC with angle bisector AD. Draw a line CE parallel to DA meeting BA extended at E. By the properties of parallel lines and similar triangles, we can establish that BD/DC = AB/AC. This theorem is useful in solving problems where we need to find unknown lengths in a triangle, especially in construction and design where precise measurements are crucial.
Discuss the concept of the area of similar triangles and how it relates to the ratio of their corresponding sides.
Recall the formula for the area of a triangle and how it scales with the sides.
Solution
The areas of similar triangles are proportional to the squares of their corresponding sides. This means if two triangles are similar with a ratio of corresponding sides k:1, then the ratio of their areas is k²:1. For example, if the sides of one triangle are twice as long as those of another similar triangle, its area will be four times as large. This concept is important in scaling objects, such as in architecture or model building, where maintaining proportions is essential.
Explain how to determine if two triangles are similar using the AA (Angle-Angle) criterion. Provide an example.
Remember that the sum of angles in a triangle is 180°.
Solution
Two triangles are similar by the AA criterion if two angles of one triangle are equal to two angles of the other triangle. Since the sum of angles in a triangle is always 180°, the third angles must also be equal. For example, if triangle ABC has angles of 50° and 60°, and triangle DEF has angles of 50° and 60°, then the third angle in both triangles is 70°, making them similar by the AA criterion. This is particularly useful in problems where only angle measures are known.
Describe the concept of the mid-segment of a triangle and its properties. How is it related to the concept of similar triangles?
Consider the properties of parallel lines and the midpoints of sides in a triangle.
Solution
The mid-segment of a triangle is the segment connecting the midpoints of two sides of the triangle. It is parallel to the third side and half as long. This creates two smaller triangles that are similar to the original triangle and to each other. For example, in triangle ABC, if D and E are the midpoints of AB and AC, respectively, then DE is parallel to BC and DE = 1/2 BC. The triangles ADE and ABC are similar by the AA criterion, as they share angle A and have corresponding angles equal due to the parallel lines.
How can the concept of similar triangles be applied to solve real-life problems, such as determining the distance across a river?
Think about creating a scale model or diagram to represent the actual scenario.
Solution
Similar triangles can be used to determine distances that are difficult to measure directly, such as the width of a river. By setting up two similar triangles where one is a small, measurable model and the other is the actual scenario, the unknown distance can be calculated. For instance, by marking two points on one side of the river and one point on the opposite side, and creating a similar triangle on land with known measurements, the width of the river can be found using the proportion of corresponding sides. This method is practical in surveying and navigation.
Triangles - Mastery Worksheet
Advance your understanding through integrative and tricky questions.
This worksheet challenges you with deeper, multi-concept long-answer questions from Triangles to prepare for higher-weightage questions in Class X.
Intermediate analysis exercises
Deepen your understanding with analytical questions about themes and characters.
Questions
Prove that the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.
Use the formula for the area of a triangle and the property of similar triangles that corresponding sides are proportional.
Solution
Let's consider two triangles ABC and DEF such that ΔABC ~ ΔDEF. By the definition of similar triangles, the corresponding sides are proportional, i.e., AB/DE = BC/EF = AC/DF = k (say). The area of triangle ABC is (1/2)*base*height, and similarly for triangle DEF. Since the triangles are similar, the heights are also in the ratio k. Therefore, the ratio of the areas is (1/2)*AB*height of ABC / (1/2)*DE*height of DEF = (AB/DE)*(height ABC/height DEF) = k*k = k², which is the square of the ratio of their corresponding sides.
In a triangle ABC, DE is parallel to BC and intersects AB and AC at D and E respectively. If AD = 4 cm, DB = 6 cm, and AE = 5 cm, find EC.
Apply the Basic Proportionality Theorem which states that a line parallel to one side of a triangle divides the other two sides proportionally.
Solution
Since DE is parallel to BC, by the Basic Proportionality Theorem (Thales' theorem), AD/DB = AE/EC. Substituting the given values, 4/6 = 5/EC. Solving for EC, we get EC = (6*5)/4 = 7.5 cm.
If the areas of two similar triangles are in the ratio 16:25, find the ratio of their corresponding sides.
Remember that the ratio of areas is the square of the ratio of corresponding sides for similar triangles.
Solution
The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides. Let the ratio of the corresponding sides be k. Then, k² = 16/25. Taking the square root of both sides, k = 4/5. Therefore, the ratio of their corresponding sides is 4:5.
In triangle ABC, angle A = 90° and AD is perpendicular to BC. Prove that AB² + AC² = BC².
Recall the Pythagorean theorem which applies to right-angled triangles.
Solution
This is a restatement of the Pythagorean theorem. In right-angled triangle ABC with angle A = 90°, by Pythagoras theorem, AB² + AC² = BC². AD being perpendicular to BC creates two smaller triangles ABD and ADC which are similar to ABC, but the Pythagorean theorem directly gives the required result.
Two triangles ABC and DEF are similar. If AB = 6 cm, DE = 4 cm, and the perimeter of ΔABC is 30 cm, find the perimeter of ΔDEF.
The ratio of the perimeters of two similar triangles is equal to the ratio of their corresponding sides.
Solution
Since ΔABC ~ ΔDEF, the ratio of their corresponding sides is equal to the ratio of their perimeters. AB/DE = 6/4 = 3/2. Therefore, the perimeter of ΔABC / perimeter of ΔDEF = 3/2. Given the perimeter of ΔABC is 30 cm, the perimeter of ΔDEF = (2/3)*30 = 20 cm.
In triangle ABC, D and E are points on AB and AC respectively such that DE is parallel to BC. If AD = 3 cm, AB = 9 cm, and AC = 12 cm, find AE and EC.
Use the Basic Proportionality Theorem to find the lengths of AE and EC.
Solution
Since DE is parallel to BC, by the Basic Proportionality Theorem, AD/AB = AE/AC. Substituting the given values, 3/9 = AE/12. Solving for AE, we get AE = (3*12)/9 = 4 cm. Therefore, EC = AC - AE = 12 - 4 = 8 cm.
Prove that in a right-angled triangle, the square on the hypotenuse is equal to the sum of the squares on the other two sides.
This is a fundamental theorem in geometry known as the Pythagorean theorem.
Solution
This is the Pythagorean theorem. Consider a right-angled triangle ABC with angle A = 90°. Construct squares on each side of the triangle. The area of the square on the hypotenuse (BC) is equal to the sum of the areas of the squares on the other two sides (AB and AC). This can be proved using similar triangles or by algebraic methods, showing that BC² = AB² + AC².
If the sides of a triangle are 6 cm, 8 cm, and 10 cm, determine whether the triangle is right-angled.
Check if the sum of the squares of the two shorter sides equals the square of the longest side.
Solution
To determine if the triangle is right-angled, we can check if it satisfies the Pythagorean theorem. The longest side is 10 cm (hypotenuse). Check if 6² + 8² = 10². 36 + 64 = 100, which is true. Therefore, the triangle is right-angled.
In triangle ABC, angle B = 90° and BD is perpendicular to AC. If AD = 4 cm and DC = 9 cm, find BD.
In a right-angled triangle, the altitude to the hypotenuse is the geometric mean of the two segments it creates on the hypotenuse.
Solution
In right-angled triangle ABC, with BD perpendicular to AC, by the property of right-angled triangles, BD² = AD*DC. Substituting the given values, BD² = 4*9 = 36. Therefore, BD = √36 = 6 cm.
Two poles of heights 6 m and 11 m stand vertically on a plane ground. If the distance between their feet is 12 m, find the distance between their tops.
Visualize the scenario as a right-angled triangle and apply the Pythagorean theorem.
Solution
The situation forms a right-angled triangle where the difference in heights is one leg (11 - 6 = 5 m), the distance between the feet is the other leg (12 m), and the distance between the tops is the hypotenuse. Using the Pythagorean theorem, distance² = 5² + 12² = 25 + 144 = 169. Therefore, the distance between their tops is √169 = 13 m.
Triangles - Challenge Worksheet
Push your limits with complex, exam-level long-form questions.
The final worksheet presents challenging long-answer questions that test your depth of understanding and exam-readiness for Triangles in Class X.
Advanced critical thinking
Test your mastery with complex questions that require critical analysis and reflection.
Questions
Prove that the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.
Consider the formula for the area of a triangle and how scaling affects both the base and height.
Solution
Use the property of similar triangles that corresponding sides are proportional and areas are proportional to the squares of the corresponding sides. Provide a step-by-step proof using the formula for the area of a triangle.
In a triangle ABC, a line DE is drawn parallel to BC, intersecting AB at D and AC at E. Prove that AD/DB = AE/EC.
Recall the conditions under which the Basic Proportionality Theorem applies.
Solution
Apply the Basic Proportionality Theorem (Thales' theorem) to show that the line divides the other two sides proportionally. Include a diagram to illustrate the scenario.
A ladder is placed against a wall such that its foot is at a distance of 2.5 m from the wall and its top reaches a window 6 m above the ground. Determine the length of the ladder.
Visualize the scenario as a right-angled triangle with the ladder as the hypotenuse.
Solution
Use the Pythagorean theorem to calculate the length of the ladder. Show the calculation step-by-step, identifying the right triangle formed by the ladder, wall, and ground.
Two triangles ABC and DEF are similar with a scale factor of 3:4. If the area of triangle ABC is 81 square units, find the area of triangle DEF.
Remember that the scale factor applies to the sides, but the area ratio is the square of this factor.
Solution
Use the property that the ratio of the areas of two similar triangles is equal to the square of the scale factor. Calculate the area of DEF accordingly.
Prove that in a right-angled triangle, the square on the hypotenuse is equal to the sum of the squares on the other two sides.
Consider constructing a perpendicular from the right angle to the hypotenuse and using properties of similar triangles.
Solution
Provide a proof of the Pythagorean theorem, possibly using similar triangles or algebraic methods. Include a diagram to support the proof.
In an equilateral triangle ABC, D is a point on BC such that BD = 1/3 BC. Prove that 9AD² = 7AB².
Drop a perpendicular from A to BC to create right triangles and apply the Pythagorean theorem.
Solution
Use the properties of equilateral triangles and the Pythagorean theorem to derive the relationship between AD and AB. Show all steps of the derivation.
A triangle has sides 7 cm, 24 cm, and 25 cm. Determine whether it is a right-angled triangle.
Identify the longest side as the potential hypotenuse and verify the Pythagorean condition.
Solution
Check if the sides satisfy the Pythagorean theorem. Calculate the squares of the sides and compare the sum of the squares of the two shorter sides to the square of the longest side.
Prove that the sum of the squares of the sides of a parallelogram is equal to the sum of the squares of its diagonals.
Consider the diagonals of the parallelogram and how they bisect each other.
Solution
Use the properties of parallelograms and the Pythagorean theorem to derive the relationship. Provide a step-by-step proof with a diagram.
In a triangle ABC, if AD is the median to side BC, prove that AB² + AC² = 2AD² + 2BD².
Recall that a median divides the opposite side into two equal parts and use the Pythagorean theorem.
Solution
Apply the Apollonius's theorem to prove the given relationship. Show the derivation step-by-step, using the properties of medians in triangles.
Two poles of heights 6 m and 11 m stand vertically on a plane ground. If the distance between their feet is 12 m, find the distance between their tops.
The difference in heights and the horizontal distance form the legs of a right triangle.
Solution
Visualize the scenario as two vertical lines with a horizontal distance between them. Use the Pythagorean theorem to find the distance between the tops of the poles.
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