Question

1. given $\triangle slz$ and $\triangle bld$. are the triangles similar? if they are, explain why. if not, explain why not.

Explanation:

Step1: List coordinates of all points

Points: $S(2,-5)$, $L(2,5)$, $Z(11,5)$, $B(2,0)$, $D(7,5)$

Step2: Calculate side lengths of $\triangle SLZ$

Use distance formula $d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$:
$SL = \sqrt{(2-2)^2+(5-(-5))^2} = \sqrt{0+100}=10$
$LZ = \sqrt{(11-2)^2+(5-5)^2} = \sqrt{81+0}=9$
$SZ = \sqrt{(11-2)^2+(5-(-5))^2} = \sqrt{81+100}=\sqrt{181}$

Step3: Calculate side lengths of $\triangle BLD$

$BL = \sqrt{(2-2)^2+(5-0)^2} = \sqrt{0+25}=5$
$LD = \sqrt{(7-2)^2+(5-5)^2} = \sqrt{25+0}=5$
$BD = \sqrt{(7-2)^2+(5-0)^2} = \sqrt{25+25}=\sqrt{50}=5\sqrt{2}$

Step4: Check side proportionality

Ratios: $\frac{SL}{BL}=\frac{10}{5}=2$, $\frac{LZ}{LD}=\frac{9}{5}=1.8$, $\frac{SZ}{BD}=\frac{\sqrt{181}}{5\sqrt{2}}\approx\frac{13.45}{7.07}\approx1.90$

Step5: Compare corresponding angles

$\triangle SLZ$ has sides 10,9,$\sqrt{181}$; $\triangle BLD$ has sides 5,5,$5\sqrt{2}$ (isosceles right triangle, one right angle, two 45° angles). $\triangle SLZ$ has no right angle ($10^2+9^2=181=(\sqrt{181})^2$, so it's a right triangle at $L$). But the other angles do not match: $\triangle BLD$ has two equal angles, $\triangle SLZ$ has all unique angles.

Answer:

The triangles are not similar. The ratios of their corresponding sides are not equal, and their corresponding angle measures do not match (only one right angle is shared, but the other angles are not congruent, and the side length proportions are inconsistent).