Prove that the triangles with the given vertices are congruent.
A(3, 1), B(4, 5), C(2, 3) and
D(–1, –3), E(–5, –4), F(–3, –2)
side AB = sqrt (16+1) = sqrt 17
side BC = sqrt (4+4) = sqrt 8
side CA = sqrt (1+4) = sqrt 5
side DE = sqrt (16+1) = sqrt 17
LOL I guess you get the SSS idea
No, I didn't get it.
hey if all three sides are the same it is congruent by side,side,
side EF = sqrt (4+4) = sqrt (8)
side FD = sqrt (1+4) = sqrt 5
the three sides of triangle 1 are the same length as the tree sides of triangle 2
To prove that two triangles are congruent, we need to show that their corresponding sides and angles are congruent. In this case, we need to compare the lengths of the sides and the measures of the angles of triangles ABC and DEF.
1. Start by finding the lengths of the sides of triangle ABC. Use the distance formula: d = sqrt((x2 - x1)^2 + (y2 - y1)^2).
- Length of side AB (point A to point B): d(AB) = sqrt((4 - 3)^2 + (5 - 1)^2) = sqrt(1^2 + 4^2) = sqrt(1 + 16) = sqrt(17).
- Length of side BC (point B to point C): d(BC) = sqrt((2 - 4)^2 + (3 - 5)^2) = sqrt((-2)^2 + (-2)^2) = sqrt(4 + 4) = sqrt(8).
- Length of side CA (point C to point A): d(CA) = sqrt((3 - 2)^2 + (1 - 3)^2) = sqrt((1)^2 + (-2)^2) = sqrt(1 + 4) = sqrt(5).
2. Find the lengths of the sides of triangle DEF using the same method:
- Length of side DE (point D to point E): d(DE) = sqrt((-5 - (-1))^2 + (-4 - (-3))^2) = sqrt((-4)^2 + (-1)^2) = sqrt(16 + 1) = sqrt(17).
- Length of side EF (point E to point F): d(EF) = sqrt((-3 - (-5))^2 + (-2 - (-4))^2) = sqrt((2)^2 + (2)^2) = sqrt(4 + 4) = sqrt(8).
- Length of side FD (point F to point D): d(FD) = sqrt((3 - (-3))^2 + (-2 - (-1))^2) = sqrt((6)^2 + (-1)^2) = sqrt(36 +1) = sqrt(37).
3. Now, compare the lengths of the sides for both triangles:
- AB = DE = sqrt(17).
- BC = EF = sqrt(8).
- CA = FD = sqrt(37).
So, all three pairs of corresponding sides are congruent.
4. Next, compare the measures of the angles in both triangles. We can use the slope formula (m = (y2 - y1) / (x2 - x1)) to find the slopes of the lines connecting the vertices of the triangles.
- Slope of line AB: m(AB) = (5 - 1) / (4 - 3) = 4 / 1 = 4.
- Slope of line BC: m(BC) = (3 - 5) / (2 - 4) = -2 / -2 = 1.
- Slope of line CA: m(CA) = (1 - 3) / (3 - 2) = -2 / 1 = -2.
- Slope of line DE: m(DE) = (-4 - (-1)) / (-5 - (-1)) = -3 / -4 = 3/4.
- Slope of line EF: m(EF) = (-2 - (-4)) / (-3 - (-5)) = 2 / 2 = 1.
- Slope of line FD: m(FD) = (-3 - (-2)) / (-1 - (-3)) = -1 / 2.
By comparing the slopes of the lines connecting the corresponding sides, we find that the slopes of AB and DE are the same (4 and 3/4), slopes of BC and EF are the same (1 and 1), and slopes of CA and FD are the same (-2 and -1/2).
Therefore, the corresponding angles are congruent as well.
Since all three pairs of corresponding sides and angles are congruent, triangles ABC and DEF are congruent by the Side-Side-Side (SSS) congruence postulate.