Question

determine the missing length in each right triangle using the pythagorean theorem. round the answer to the nearest tenth.
1)
triangle abc with right angle at b, ab = 12 yd, ac = 15 yd, find bc.
2)
triangle pqr with right angle at r, qr = 2 in, pr = 5 in, find pq.
3)
triangle uvw with right angle at u, uv = 6 ft, vw = 9 ft, find uw.
4)
triangle xyz with right angle at y, yz = 14 in, xy = 3 in, find xz.
5)
triangle efg with right angle at f, fg = 8 ft, eg = 10 ft, find ef.
6)
triangle jkl with right angle at k, jk = 13 yd, kl = 17 yd, find jl.
7)
triangle lmn with right angle at l, lm = 8 ft, ln = 16 ft, find mn.
8)
triangle stu with right angle at t, tu = 1 yd, su = 4 yd, find st.
9)
triangle cde with right angle at d, cd = 7 in, de = 11 in, find ce.

Explanation:

Step1: Recall Pythagorean Theorem

The Pythagorean Theorem states that for a right triangle with legs \(a\) and \(b\), and hypotenuse \(c\), \(a^{2}+b^{2}=c^{2}\). We will apply this to each triangle.

Problem 1: Triangle \(ABC\) (12 yd, 15 yd, find \(BC\))

Step1: Identify sides

Leg \(AB = 12\) yd, hypotenuse \(AC=15\) yd, leg \(BC =?\)

Step2: Apply Pythagorean Theorem

\(AB^{2}+BC^{2}=AC^{2}\)
\(12^{2}+BC^{2}=15^{2}\)
\(144 + BC^{2}=225\)
\(BC^{2}=225 - 144=81\)
\(BC=\sqrt{81} = 9\) yd

Problem 2: Triangle \(PQR\) (2 in, 5 in, find \(PQ\))

Step1: Identify sides

Leg \(QR = 2\) in, leg \(PR = 5\) in, hypotenuse \(PQ=?\)

Step2: Apply Pythagorean Theorem

\(QR^{2}+PR^{2}=PQ^{2}\)
\(2^{2}+5^{2}=PQ^{2}\)
\(4 + 25=PQ^{2}\)
\(PQ^{2}=29\)
\(PQ=\sqrt{29}\approx5.4\) in

Problem 3: Triangle \(UVW\) (6 ft, 9 ft, find \(UW\))

Step1: Identify sides

Leg \(UV = 6\) ft, hypotenuse \(VW = 9\) ft, leg \(UW=?\)

Step2: Apply Pythagorean Theorem

\(UV^{2}+UW^{2}=VW^{2}\)
\(6^{2}+UW^{2}=9^{2}\)
\(36+UW^{2}=81\)
\(UW^{2}=81 - 36 = 45\)
\(UW=\sqrt{45}\approx6.7\) ft

Problem 4: Triangle \(XYZ\) (14 in, 3 in, find \(XZ\))

Step1: Identify sides

Leg \(YZ = 14\) in, leg \(XY = 3\) in, hypotenuse \(XZ=?\)

Step2: Apply Pythagorean Theorem

\(YZ^{2}+XY^{2}=XZ^{2}\)
\(14^{2}+3^{2}=XZ^{2}\)
\(196 + 9=XZ^{2}\)
\(XZ^{2}=205\)
\(XZ=\sqrt{205}\approx14.3\) in

Problem 5: Triangle \(EFG\) (8 ft, 10 ft, find \(EF\))

Step1: Identify sides

Leg \(FG = 8\) ft, hypotenuse \(EG = 10\) ft, leg \(EF=?\)

Step2: Apply Pythagorean Theorem

\(EF^{2}+FG^{2}=EG^{2}\)
\(EF^{2}+8^{2}=10^{2}\)
\(EF^{2}+64 = 100\)
\(EF^{2}=100 - 64=36\)
\(EF=\sqrt{36}=6\) ft

Problem 6: Triangle \(JKL\) (13 yd, 17 yd, find \(JL\))

Step1: Identify sides

Leg \(JK = 13\) yd, leg \(KL = 17\) yd, hypotenuse \(JL=?\)

Step2: Apply Pythagorean Theorem

\(JK^{2}+KL^{2}=JL^{2}\)
\(13^{2}+17^{2}=JL^{2}\)
\(169+289 = JL^{2}\)
\(JL^{2}=458\)
\(JL=\sqrt{458}\approx21.4\) yd

Problem 7: Triangle \(LMN\) (8 ft, 16 ft, find \(MN\))

Step1: Identify sides

Leg \(LM = 8\) ft, leg \(LN = 16\) ft, hypotenuse \(MN=?\)

Step2: Apply Pythagorean Theorem

\(LM^{2}+LN^{2}=MN^{2}\)
\(8^{2}+16^{2}=MN^{2}\)
\(64 + 256=MN^{2}\)
\(MN^{2}=320\)
\(MN=\sqrt{320}\approx17.9\) ft

Problem 8: Triangle \(STU\) (1 yd, 4 yd, find \(ST\))

Step1: Identify sides

Leg \(TU = 1\) yd, hypotenuse \(SU = 4\) yd, leg \(ST=?\)

Step2: Apply Pythagorean Theorem

\(ST^{2}+TU^{2}=SU^{2}\)
\(ST^{2}+1^{2}=4^{2}\)
\(ST^{2}+1 = 16\)
\(ST^{2}=15\)
\(ST=\sqrt{15}\approx3.9\) yd

Problem 9: Triangle \(CDE\) (7 in, 11 in, find \(CE\))

Step1: Identify sides

Leg \(CD = 7\) in, leg \(DE = 11\) in, hypotenuse \(CE=?\)

Step2: Apply Pythagorean Theorem

\(CD^{2}+DE^{2}=CE^{2}\)
\(7^{2}+11^{2}=CE^{2}\)
\(49+121 = CE^{2}\)
\(CE^{2}=170\)
\(CE=\sqrt{170}\approx13.0\) in

Answer:

1. \(BC = \boldsymbol{9}\) yd
2. \(PQ\approx\boldsymbol{5.4}\) in
3. \(UW\approx\boldsymbol{6.7}\) ft
4. \(XZ\approx\boldsymbol{14.3}\) in
5. \(EF = \boldsymbol{6}\) ft
6. \(JL\approx\boldsymbol{21.4}\) yd
7. \(MN\approx\boldsymbol{17.9}\) ft
8. \(ST\approx\boldsymbol{3.9}\) yd
9. \(CE\approx\boldsymbol{13.0}\) in