Question

use efficient methods determine whether △def ≅ △pqr. d(-6, 1), e(1, 2), f(-1, -4), p(0, 5), q(7, 6), r(5, 0) find the side lengths of each triangle. a) de = 5√2, pq = 5√2, ef = √10, qr = √10, df = 5√2, pr = 5√2 b) de = √2, pq = √2, ef = √10, qr = √10, df = √2, pr = √2 c) de = 5√2, pq = 5√2, ef = 2√10, qr = 2√10, df = 5√2, pr = 5√2 d) de = 5√2, pq = 5√10, ef = 2√10, qr = 2√10, df = 5√10, pr = 5√2 need help?

Explanation:

Step1: Recall Distance Formula

The distance between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \).

Step2: Calculate \( DE \) (D(-6,1), E(1,2))

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Step3: Calculate \( PQ \) (P(0,5), Q(7,6))

$$
LATEXBLOCK1
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Step4: Calculate \( EF \) (E(1,2), F(-1,-4))

$$
LATEXBLOCK2
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Step5: Calculate \( QR \) (Q(7,6), R(5,0))

$$
LATEXBLOCK3
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Step6: Calculate \( DF \) (D(-6,1), F(-1,-4))

$$
LATEXBLOCK4
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Step7: Calculate \( PR \) (P(0,5), R(5,0))

$$
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$$

By comparing the side lengths: \( DE = PQ = 5\sqrt{2} \), \( EF = QR = 2\sqrt{10} \), \( DF = PR = 5\sqrt{2} \), so the correct option is C.

Answer:

C) \( DE = 5\sqrt{2} \), \( PQ = 5\sqrt{2} \), \( EF = 2\sqrt{10} \), \( QR = 2\sqrt{10} \), \( DF = 5\sqrt{2} \), \( PR = 5\sqrt{2} \)