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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).
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.
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.
Describe the criteria for similarity of triangles and give an example for each criterion.
Recall the conditions under which two triangles are considered 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.
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.
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.
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.
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°.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
Real Numbers encompass all rational and irrational numbers, forming a complete and continuous number line essential for various mathematical concepts.
Explore the world of Polynomials, understanding their types, degrees, and operations to solve algebraic expressions and equations effectively.
Explore the methods to solve a pair of linear equations in two variables, including graphical, substitution, elimination, and cross-multiplication techniques.
Explore the world of quadratic equations, learning to solve them using various methods like factoring, completing the square, and the quadratic formula.
A chapter that explores sequences where each term after the first is obtained by adding a constant difference, focusing on their properties, nth term, and sum formulas.
