Lesson 2

Half a Square

The practice problem answers are available at one of our IM Certified Partners

Problem 1

Find the lengths of the legs.

Triangle. An angle is marked as a right angle. Another angle is labeled 45 degrees. The side opposite the right angle is labeled 4.

$4\sqrt{2}$ units


$\frac{4}{\sqrt{2}}$ units


4 units


Not enough information

Problem 2

What is the length of the diagonal?

A four sided figure. Each side has a single tick mark. Each angle is marked as a right angle. The top of the figure is labeled 1.

Problem 3

A square has a diagonal of length 5 cm. What is the area of the square?

Problem 4

Priya is teaching her younger cousin to ride a bike. She wants to stay on roads that are not too steep and easy enough for a new bike rider. She has decided the roads must have an angle less than or equal to 7 degrees. A 7 degree angle in a right triangle has a \(3:25\) ratio for the legs. List the legs of 2 right triangles that would be safe for a new bike rider. 

(From Geometry, Unit 4, Lesson 1.)

Problem 5

Clare and Han are discussing how to find the missing lengths. Clare says she is using similarity. Han says he is using the Pythagorean Theorem.

  1. Do you agree with either of them? Show or explain your reasoning. 
  2. Find the missing sides. 
Two right triangles. Triangle A B C has right angle B, side A B is 2 point 5 and B C is 6. Triangle D E F has right angle E, side E F is 5 and D F is 13.
(From Geometry, Unit 4, Lesson 1.)

Problem 6

In right triangle \(ABC\), angle \(C\) is a right angle, \(AB\) is 25 units long, and \(BC\) is 24 units long. What is the length of \(AC\)?









(From Geometry, Unit 3, Lesson 15.)

Problem 7

  1. Find the length of \(EF\).
  2. Find the measure of angle \(E\).
Two triangles, A B C and D E F. Angle and side labels as follows: Angle A B C, 82. Angle B A C, 55. Side B C, 12. Side A C, 14. Angle D F E, 43. Angle E D F, 55. Side D F, 21.
(From Geometry, Unit 3, Lesson 10.)

Problem 8

Determine if each statement must be true, could possibly be true, or definitely can't be true. Explain or show your reasoning.

  1. Isosceles triangles are similar. 
  2. Equilateral triangles are similar. 
(From Geometry, Unit 3, Lesson 7.)