Question

the sides \\( \overline { l m } \\), \\( \overline { m n } \\), and \\( \overline { l n } \\) of \\( \delta l m n \\) have sides of length \\( 15 \sqrt { 2 } \\), 15, and 15, respectively. which triangle is similar to \\( \delta l m n \\) using the sss similarity theorem?

Explanation:

Step1: Check the side - length ratios

Let's assume the side - lengths of \(\triangle LMN\) are \(a = 15\), \(b = 15\), \(c=15\sqrt{2}\).
We know that for a right - isosceles triangle with legs \(x\) and hypotenuse \(y\), by the Pythagorean theorem \(y=\sqrt{x^{2}+x^{2}}=\sqrt{2x^{2}}=\sqrt{2}x\).
If we consider a smaller right - isosceles triangle with legs of length \(l\) and hypotenuse \(h\), we want to find the ratio of the sides of \(\triangle LMN\) to the sides of the candidate triangles.
Let's use the distance formula \(d=\sqrt{(x_2 - x_1)^{2}+(y_2 - y_1)^{2}}\) to find the side - lengths of the triangles in the coordinate - plane.
For the first triangle (left - most):
Let's assume two vertices of a right - angled side: if we consider two points \((- 3,1)\) and \((-1,3)\), the length of the side \(d_1=\sqrt{(-1 + 3)^{2}+(3 - 1)^{2}}=\sqrt{4 + 4}=\sqrt{8}=2\sqrt{2}\), another right - angled side (e.g., from \((-3,1)\) to \((-3,3)\)) \(d_2 = 2\), and the hypotenuse (from \((-3,3)\) to \((-1,3)\)) \(d_3=\sqrt{(-1 + 3)^{2}+(3 - 3)^{2}}=2\) (incorrect, not a right - isosceles triangle by Pythagorean check \(d_1^{2}
eq d_2^{2}+d_3^{2}\)).
For the second triangle:
Using the distance formula, if we consider two points \((-1,3)\) and \((1, - 3)\), \(d=\sqrt{(1 + 1)^{2}+(-3 - 3)^{2}}=\sqrt{4 + 36}=\sqrt{40}=2\sqrt{10}\), another side (e.g., from \((-1,3)\) to \((-1,-4)\)) \(d_1 = 7\), and from \((-1,-4)\) to \((1,-4)\) \(d_2 = 2\) (not a right - isosceles triangle).
For the third triangle:
Let's assume two vertices on the \(x\) - axis \((-3,0)\) and \((3,0)\), the length of this side \(d_1=6\), and the length from \((0,4)\) to \((-3,0)\) \(d_2=\sqrt{(0 + 3)^{2}+(4 - 0)^{2}}=\sqrt{9 + 16}=5\), from \((0,4)\) to \((3,0)\) \(d_3=\sqrt{(0 - 3)^{2}+(4 - 0)^{2}}=5\) (not a right - isosceles triangle).
For the fourth triangle:
Let's assume two vertices: if we consider two points \((-2,3)\) and \((-2,-3)\), the length of this side \(d_1 = 6\), another side (e.g., from \((-2,-3)\) to \((2,-1)\)):
\(d_2=\sqrt{(2 + 2)^{2}+(-1 + 3)^{2}}=\sqrt{16 + 4}=\sqrt{20}=2\sqrt{5}\), and the hypotenuse (from \((-2,3)\) to \((2,-1)\)):
\(d_3=\sqrt{(2 + 2)^{2}+(-1 - 3)^{2}}=\sqrt{16 + 16}=\sqrt{32}=4\sqrt{2}\).
If we scale down \(\triangle LMN\) (sides \(15\), \(15\), \(15\sqrt{2}\)) by a factor of \(\frac{1}{ \frac{15}{4}}\) (i.e., consider the ratio of sides).
Let's assume a right - isosceles triangle with legs \(l\) and hypotenuse \(h\). If we consider a triangle with legs of length \(4\) (e.g., from \((-2,3)\) to \((-2,-1)\)) \(l = 4\), another leg (from \((-2,-1)\) to \((2,-1)\)) \(l = 4\) (after adjusting the coordinate - based side - length calculation using the distance formula \(d=\sqrt{(x_2 - x_1)^{2}+(y_2 - y_1)^{2}}\), for two points \((x_1,y_1)\) and \((x_2,y_2)\)). The hypotenuse (from \((-2,3)\) to \((2,-1)\)) \(d=\sqrt{(2 + 2)^{2}+(-1 - 3)^{2}}=\sqrt{16 + 16}=4\sqrt{2}\).
The ratio of the sides of \(\triangle LMN\) (\(15,15,15\sqrt{2}\)) to the sides of the fourth triangle (legs \(4\) and \(4\), hypotenuse \(4\sqrt{2}\)) is \(\frac{15}{4}\).

Answer:

The fourth triangle (right - most) is similar to \(\triangle LMN\) by the SSS (Side - Side - Side) similarity theorem.