Given: A(3,-1), B(5,2), C(-2,0), P(-3,4), Q(-5,-3), R(-6,2).
Prove: angles ABC and RPQ are congruent by completing the paragraph proof.
AB=RP=13, BC=(?)=53, and CA=QR=26. So segment AB is congruent to (?), segments BC and PQ are congruent and segment CA is congruent to segment QR. Therefore triangle ABC is congruent to (?) by (?), and angles ABC and RPQ are congruent by (?).
AB = √[(5-3)^2 + (2+1)^2] = √(4+9) = √13
RP = √([-3+6)^2 +(4-2)^2] = √(9+4) = √13
BC = √[(5+2)^2 + (2-0)^2] = √(49+4) = √53
PQ = √[-3+5)^2 + (4+3)^2] = √(4+49) = √53
AC = √[3+2)^2 + (-1-0)^2] = √(25+1) = √26
RQ = √[(-6+5)^2 + (2+3)^2] = √(1+25) = √26
clearly we have corresponding pairs of sides equal, so
by SSS, ∆ABC≅∆RPQ
(Your length of 13, 53, and 26 should have been √13, √53, and √26)
I guess I don't understand how to write the answer.
Angle ABC is congruent to RPQ by____? and angles ABC and RPQ are congruent by _____?
To prove that angles ABC and RPQ are congruent, we need to show that triangle ABC is congruent to triangle RPQ.
First, let's find the length of BC. To find the length of BC, we can use the distance formula:
d = √((x2 - x1)^2 + (y2 - y1)^2)
Using points B(5,2) and C(-2,0):
BC = √((5 - (-2))^2 + (2 - 0)^2)
= √((7)^2 + (2)^2)
= √(49 + 4)
= √53
Therefore, BC = √53.
Next, let's find the length of QR. Using points Q(-5,-3) and R(-6,2):
QR = √((-5 - (-6))^2 + (-3 - 2)^2)
= √((-5 + 6)^2 + (-3 - 2)^2)
= √((1)^2 + (-5)^2)
= √(1 + 25)
= √26
Therefore, QR = √26.
Now that we have the lengths of the sides of triangle ABC and triangle RPQ, we can use the Side-Side-Side (SSS) congruence criterion to prove that the triangles are congruent.
We know that AB = RP = 13 (given), BC = √53 (calculated), and CA = QR = √26 (calculated).
Therefore, triangle ABC is congruent to triangle RPQ by SSS.
Since corresponding sides in congruent triangles are congruent, we can conclude that angle ABC and angle RPQ are congruent.
So, the missing values are:
- BC = √53
- triangle ABC is congruent to triangle RPQ
- angles ABC and RPQ are congruent