To find the missing lengths in these right triangles, we use the Pythagorean Theorem. This rule says that for any right triangle:
$$a^2 + b^2 = c^2$$
*
$a$ and $b$ are the two shorter sides (the legs).
*
$c$ is the longest side (the hypotenuse), which is always opposite the right angle.
Here is the step-by-step solution for each problem on the worksheet:
1. Find BC
* Legs: 5 and 12. Hypotenuse: BC ($c$)
* $5^2 + 12^2 = c^2$
* $25 + 144 = 169$
* $\sqrt{169} = 13$
*
BC = 13
2. Find PQ
* Legs: 8 and 15. Hypotenuse: PQ ($c$)
* $8^2 + 15^2 = c^2$
* $64 + 225 = 289$
* $\sqrt{289} = 17$
*
PQ = 17
3. Find MN
* Legs: 7 and 24. Hypotenuse: MN ($c$)
* $7^2 + 24^2 = c^2$
* $49 + 576 = 625$
* $\sqrt{625} = 25$
*
MN = 25
4. Find XZ
* Legs: 9 and 12. Hypotenuse: XZ ($c$)
* $9^2 + 12^2 = c^2$
* $81 + 144 = 225$
* $\sqrt{225} = 15$
*
XZ = 15
5. Find XY
* Leg: 5. Hypotenuse: 13. Missing Leg: XY ($a$)
* $a^2 + 5^2 = 13^2$
* $a^2 + 25 = 169$
* Subtract 25 from both sides: $a^2 = 144$
* $\sqrt{144} = 12$
*
XY = 12
6. Find JK
* Leg: 8. Hypotenuse: 17. Missing Leg: JK ($a$)
* $a^2 + 8^2 = 17^2$
* $a^2 + 64 = 289$
* Subtract 64 from both sides: $a^2 = 225$
* $\sqrt{225} = 15$
*
JK = 15
7. Find EF
* Leg: 5. Hypotenuse: 13. Missing Leg: EF ($a$)
* $a^2 + 5^2 = 13^2$
* $a^2 + 25 = 169$
* Subtract 25 from both sides: $a^2 = 144$
* $\sqrt{144} = 12$
*
EF = 12
8. Find ST
* Leg: 8. Hypotenuse: 10. Missing Leg: ST ($a$)
* $a^2 + 8^2 = 10^2$
* $a^2 + 64 = 100$
* Subtract 64 from both sides: $a^2 = 36$
* $\sqrt{36} = 6$
*
ST = 6
9. Find CD
* Legs: 9 and 12. Hypotenuse: CD ($c$)
* $9^2 + 12^2 = c^2$
* $81 + 144 = 225$
* $\sqrt{225} = 15$
*
CD = 15
Final Answer:
1. BC = 13
2. PQ = 17
3. MN = 25
4. XZ = 15
5. XY = 12
6. JK = 15
7. EF = 12
8. ST = 6
9. CD = 15
Parent Tip: Review the logic above to help your child master the concept of pythagorean theorem worksheet answer key.
