Question

pythagorean theorem - worksheet
for each triangle find the missing length. round your answer to the nearest tenth. then find the area and the perimeter.
1.
triangle with sides 19, 17, and x
2.
right triangle with legs 5, 13, and hypotenuse?
3.
right triangle with hypotenuse 20, leg 10, and leg?
4.
right triangle with leg 5, leg 14, and hypotenuse?

Explanation:

Problem 1:

Step 1: Identify triangle type and formula

This is a right triangle (implied by Pythagorean theorem use). Let the hypotenuse be \( c = 19 \), one leg \( b = 17 \), and the other leg \( a = x \). The Pythagorean theorem is \( a^{2}+b^{2}=c^{2} \), so we solve for \( a \): \( a=\sqrt{c^{2}-b^{2}} \)

Step 2: Substitute values

Substitute \( c = 19 \) and \( b = 17 \) into the formula: \( x=\sqrt{19^{2}-17^{2}}=\sqrt{(19 + 17)(19 - 17)}=\sqrt{36\times2}=\sqrt{72}\approx8.5 \)

Step 3: Calculate area

Area of a right triangle is \( A=\frac{1}{2}\times\text{leg}_1\times\text{leg}_2=\frac{1}{2}\times17\times8.5 = 72.25\approx72.3 \)

Step 4: Calculate perimeter

Perimeter \( P=17 + 19+8.5 = 44.5 \)

Step 1: Identify triangle type and formula

Right triangle, hypotenuse \( c =? \), legs \( a = 5 \), \( b = 13 \). Use Pythagorean theorem \( c=\sqrt{a^{2}+b^{2}} \)

Step 2: Substitute values

\( c=\sqrt{5^{2}+13^{2}}=\sqrt{25 + 169}=\sqrt{194}\approx13.9 \)

Step 3: Calculate area

Area \( A=\frac{1}{2}\times5\times13 = 32.5 \)

Step 4: Calculate perimeter

Perimeter \( P=5 + 13+13.9 = 31.9 \)

Step 1: Identify triangle type and formula

Right triangle, hypotenuse \( c = 20 \), leg \( b = 10 \), leg \( a =? \). Use \( a=\sqrt{c^{2}-b^{2}} \)

Step 2: Substitute values

\( a=\sqrt{20^{2}-10^{2}}=\sqrt{400 - 100}=\sqrt{300}\approx17.3 \)

Step 3: Calculate area

Area \( A=\frac{1}{2}\times10\times17.3 = 86.5 \)

Step 4: Calculate perimeter

Perimeter \( P=10 + 20+17.3 = 47.3 \)

Answer:

Missing length \( x\approx8.5 \), Area \( \approx72.3 \), Perimeter \( = 44.5 \)

Problem 2: