Check Your Skills Answer Key

1. No

   If the two known sides of the triangle are 5 and 19, then the third side of the triangle cannot have a length of 13, because that would violate the rule that any two sides of a triangle must add up to greater than the third side: 5 + 13 = 18, and 18 < 19:

   No possible triangle with these lengths.

2. 9 < third side < 25

   If the two known sides of the triangle are 8 and 17, then the third side must be less than the sum of the other two sides: 8 + 17 = 25, so the third side must be less than 25. The third side must also be greater than the difference of the other two sides: 17 − 8 = 9, so the third side must be greater than 9. That means that 9 < third side < 25.

3. 65°

   The internal angles of a triangle must add up to 180°, so you know that 40 + 75 + x = 180. Solving for x gives you x = 65°:

   

4. 65°

   The three internal angles of the triangle must add up to 180°, so 50 + x + x = 180. That means that 2x = 130, and x = 65:

   

5. x = 70°, y=80°

   To determine the missing angles of the triangle, you need to do a little work with the picture. You can figure out the value of x, because straight lines have a degree measure of 180, so 110 + x = 180, which means x = 70. That means your picture looks like this:

   

   Now you can find y, because 30 + 70 + y = 180. Solving for y gives you y = 80:

   

6. 80°

   In this triangle, two sides have the same length, which means this triangle is isosceles. You also know that the two angles opposite the two equal sides will also be equal. That means that x must be 80:

   

7. 4

   In this triangle, two angles are equal, which means this triangle is isosceles. Thus, you also know that the two sides opposite the equal angles must also be equal, so x must equal 4:

   

8. 110°

   This triangle is isosceles, because two sides have the same length. That means the angles opposite the equal sides must also be equal. That means the triangle really looks like this:

   

   Now you can find x, because you know 35 + 35 + x = 180. Solving for x gives you x = 110:

   

9. 25

   

   To find the perimeter of the triangle, add up all three sides: 5 + 8 + 12 = 25. Thus, the perimeter is 25.

10. 16

    To find the perimeter of the triangle, you need the lengths of all three sides. This is an isosceles triangle, because two angles are equal. That means that the sides opposite the equal angles must also be equal. So your triangle looks like this:

    

    The perimeter is 6 + 6 + 4, which equals 16.

11. 15

    The area of a triangle is . In the triangle shown, the base is 6 and the height is 5, so the area is , which equals 15.

12. 35

    In this triangle, the base is 10 and the height is 7. Remember that the height must be perpendicular to the base—it doesn’t need to lie within the triangle. Thus, the area is , which equals 35.

13. 6

    This is a right triangle, so you can use the Pythagorean theorem to solve for the length of the third side. The hypotenuse is the side with length 10, so the formula is (8)² + b² = (10)². Thus, 64 + b² = 100, so b² is 36, which means b = 6. The third side of the triangle has a length of 6. Alternatively, you could recognize that this triangle is one of the Pythagorean triples—a 6–8–10 triangle, which is just a doubled 3–4–5 triangle.

14. 13

    This is a right triangle, so you can use the Pythagorean theorem to solve for the length of the third side. The hypotenuse is the unknown side, so the formula is (5)² + (12)² = c². Thus, 25 + 144 = c², so c²  = 169, which means c is 13. The third side of the triangle has a length of 13. Alternatively, you could recognize that this triangle is one of the Pythagorean triples—a 5–12–13 triangle.

15. 6

    This is a right triangle, so you can use the Pythagorean theorem to solve for the third side, or recognize that this is a 3–4–5 triangle. Either way, the result is the same: the length of the third side is 3:

    

    Now you can find the area of the triangle. Area of a triangle is , so the area of this triangle is , which equals 6.

16. 

    Apply the Pythagorean theorem directly, substituting 1 for a and b, and C for c:

    
17. 

    Apply the Pythagorean theorem directly, substituting 1 for a and 2 for c, and B for b:

    
18. 3 or 4

    Because an isosceles triangle has two equal sides, the third side must be equal to one of the two named sides.

19. 18

    Call the side length of the square x. Thus, the diagonal would be . You know the diagonal is 6, so . This means . The area is x × x, or .

20. 

    If the area is 25, the side length x is 5. Because the diagonal is , the diagonal is .

21. 40

    The long diagonal BD is the sum of two long legs of the 30–60–90 triangle, so each long leg is . The leg:leg:hypotenuse ratio of a 30–60–90 triangle is , which means that . Therefore, x = 5, so the length of the short leg is 5 and the length of the hypotenuse is 10. Because the perimeter of the figure is the sum of four hypotenuses, the perimeter of this figure is 40.

22. 

    The line along which the height is measured in the figure bisects the equilateral triangle, creating two identical 30–60–90 triangles, each with a base of 1. The base of each of these triangles is the short leg of a 30–60–90 triangle. Because the leg:leg:hypotenuse ratio of a 30–60–90 triangle is , the long leg of each 30–60–90 triangle, also the height of the equilateral triangle, is :

    

23. 26

    The diagonal of the rectangle is the hypotenuse of a right triangle whose legs are the length and width of the rectangle. In this case, that means that the legs of the right triangle are 10 and 24. Plug these leg lengths into the Pythagorean theorem:

    You could use the calculator to take this big square root.

    Alternatively, you could recognize the 10 : 24 : 26 triangle (a multiple of the more common 5 : 12 : 13 triangle) and save yourself the trouble.

24. 

    The perimeter of a rectangle is 2(length + width). In this case, that means 2(x + 2x), which equals 6x. You are told the perimeter equals 6, so 6x = 6, and x is 1. Therefore, the length (2x) is 2 and the width (x) is 1. The diagonal of the rectangle is the hypotenuse of a right triangle whose legs are the length and width of the rectangle. Plug the leg lengths into the Pythagorean theorem: