Question

18
congruent by: answer
δcan~δ answer (if they are not similar write xxx)

Explanation:

Step1: Check for Congruence

For two right - angled triangles to be congruent, we can use the Hypotenuse - Leg (HL) criterion. In \(\triangle ACN\) and \(\triangle VHL\), we need to check the lengths of the hypotenuse and one leg.
The hypotenuse of \(\triangle ACN\): Using the Pythagorean theorem \(c=\sqrt{a^{2}+b^{2}}\), where \(a = 48\) and \(b=51\). So, \(CN=\sqrt{48^{2}+51^{2}}=\sqrt{2304 + 2601}=\sqrt{4905}\).
The hypotenuse of \(\triangle VHL\): Using the Pythagorean theorem \(c=\sqrt{a^{2}+b^{2}}\), where \(a = 44\) and \(b = 46.75\). So, \(VH=\sqrt{44^{2}+46.75^{2}}=\sqrt{1936+2185.5625}=\sqrt{4121.5625}=64.2\) (approx). Since the hypotenuses are not equal and the legs \(48
eq44\) and \(51
eq46.75\), the triangles are not congruent.

Step2: Check for Similarity (using the ratio of sides)

For similarity of right - angled triangles (by SAS - similarity for right - angled triangles, we check the ratio of the legs.
The ratio of the first pair of legs: \(\frac{48}{44}=\frac{12}{11}\approx1.09\)
The ratio of the second pair of legs: \(\frac{51}{46.75}=\frac{51\times4}{46.75\times4}=\frac{204}{187}=\frac{12}{11}\approx1.09\)
Since the ratio of the corresponding legs of the two right - angled triangles \(\triangle ACN\) and \(\triangle VHL\) are equal (\(\frac{AC}{VL}=\frac{AN}{HL}=\frac{12}{11}\)) and the included angles (right angles) are equal.

Answer:

\(\triangle CAN\sim\triangle VHL\) by \(SAS\sim\)