Question 1: An isosceles triangle has two equal sides of length 7 cm. The base is 10 cm. Calculate the height drawn to the base.

Answer: 6.4

Question 2: Construct an isosceles triangle with equal sides of 6 cm and a base of 8 cm on the grid. Label the sides.

Answer: Construction on grid

Question 3: In an isosceles triangle, the vertex angle measures 50°. Find the measures of the base angles.

Answer: 65° each

Question 4: Plot the graph of y=3x + 2 and draw an isosceles triangle with its base on the line y=2 and vertex on y=8.

Answer: Graph and triangle drawn on grid

Question 5: A ladder leaning against a wall forms an isosceles triangle with the ground and the wall. If the ladder is 10 meters long and touches the wall at 8 meters height, what is the distance from the wall to the ladder's base?

Answer: 6 meters

Question 6: A student claims that all isosceles triangles are also equilateral. Is this statement true or false? Explain your reasoning.

Answer: False. Isosceles triangles have two equal sides, but not necessarily all three, which is required for equilateral triangles.

Question 7: Find the length of the equal sides of an isosceles triangle if the base is 5 cm, and the height from the vertex to the base is 4 cm.

Answer: 6.4 cm

Question 8: In an isosceles triangle, the two equal sides are 9 cm each, and the vertex angle is 60°. Calculate the base angles.

Answer: 60° each

Question 9: Identify and correct the mistake: 'An isosceles triangle with two equal sides must have equal base angles, but in the diagram, the angles are different.'

Answer: The mistake is that in an isosceles triangle, the angles opposite the equal sides are equal; if shown differently, it's incorrect.

Question 10: Prove that the base angles of an isosceles triangle with vertex angle 40° are equal.

Answer: Since the sum of angles in a triangle is 180°, the base angles each measure (180° – 40°)/2 = 70°, and are equal by the properties of isosceles triangles.

Question 11: A real-world example: A traffic sign is an isosceles triangle with two sides of 12 inches and a base of 16 inches. Calculate the height of the sign.

Answer: 9 inches

Question 12: Challenge: Given an isosceles triangle with sides 8 cm, 8 cm, and a base of x cm. Derive an expression for the height in terms of x, and find the height when x = 6 cm.

Answer: Height = √(64 – (x²/4)); for x=6, height = √(64 – 9) = √55 ≈ 7.42 cm