Chapter 4, Problem 12PT

To determine

To explain:

  $△\mathrm{T}\mathrm{J}\mathrm{D}\cong △\mathrm{S}\mathrm{E}\mathrm{K}$

Answer to Problem 12PT

Corresponding sides are congruent one obtains $△\mathrm{T}\mathrm{J}\mathrm{D}\cong △\mathrm{S}\mathrm{E}\mathrm{K}$

Explanation of Solution

Given information:

  $\mathrm{T}\left(-4,-2\right),\mathrm{J}\left(0,5\right),\mathrm{D}\left(1,-1\right),\mathrm{S}\left(-1,3\right),\mathrm{E}\left(3,10\right),\mathrm{K}\left(4,4\right)$

Find the length of each side of $△\mathrm{T}\mathrm{J}\mathrm{D}$ .

Find the length of $\overline{\mathrm{T}\mathrm{J}}$ with $\mathrm{T}\left(-4,-2\right),\mathrm{J}\left(0,5\right)$

  $\begin{array}{l}\mathrm{T}\mathrm{J}=\sqrt{{\left({\mathrm{x}}_2-{\mathrm{x}}_1\right)}^2+{\left({\mathrm{y}}_2-{\mathrm{y}}_1\right)}^2}\\ \mathrm{T}\mathrm{J}=\sqrt{{\left(0-\left(-4\right)\right)}^2+{\left(5-\left(-2\right)\right)}^2}\\ \mathrm{T}\mathrm{J}=\sqrt{16+49}\\ \mathrm{T}\mathrm{J}=\sqrt{65}\end{array}$

Find the length of $\overline{\mathrm{J}\mathrm{D}}$ with $\mathrm{T}\left(-4,-2\right),\mathrm{D}\left(1,-1\right)$

  $\begin{array}{l}\mathrm{J}\mathrm{D}=\sqrt{{\left({\mathrm{x}}_2-{\mathrm{x}}_1\right)}^2+{\left({\mathrm{y}}_2-{\mathrm{y}}_1\right)}^2}\\ \mathrm{J}\mathrm{D}=\sqrt{{\left(1-0\right)}^2+{\left(-1-\left(5\right)\right)}^2}\\ \mathrm{J}\mathrm{D}=\sqrt{1+\left(36\right)}\\ \mathrm{J}\mathrm{D}=\sqrt{37}\\ \mathrm{J}\mathrm{D}=\sqrt{5}\end{array}$

Find the length of $\overline{\mathrm{T}\mathrm{D}}$ with $\mathrm{J}\left(0,5\right),\mathrm{D}\left(1,-1\right)$

  $\begin{array}{l}\mathrm{T}\mathrm{D}=\sqrt{{\left({\mathrm{x}}_2-{\mathrm{x}}_1\right)}^2+{\left({\mathrm{y}}_2-{\mathrm{y}}_1\right)}^2}\\ \mathrm{T}\mathrm{D}=\sqrt{{\left(1-\left(-4\right)\right)}^2+{\left(-1-\left(-2\right)\right)}^2}\\ \mathrm{T}\mathrm{D}=\sqrt{25+\left(1\right)}\\ \mathrm{T}\mathrm{D}=\sqrt{26}\end{array}$

Second find the length of each side $△\mathrm{S}\mathrm{E}\mathrm{K}$ ,

Find the length of $\overline{\mathrm{S}\mathrm{E}}$ with $\mathrm{S}\left(-1,3\right),\mathrm{E}\left(3,10\right)$

  $\begin{array}{l}\mathrm{S}\mathrm{E}=\sqrt{{\left({\mathrm{x}}_2-{\mathrm{x}}_1\right)}^2+{\left({\mathrm{y}}_2-{\mathrm{y}}_1\right)}^2}\\ \mathrm{S}\mathrm{E}=\sqrt{{\left(3-\left(-1\right)\right)}^2+{\left(10-3\right)}^2}\\ \mathrm{S}\mathrm{E}=\sqrt{16+\left(49\right)}\\ \mathrm{S}\mathrm{E}=\sqrt{65}\end{array}$

Find the length of $\overline{\mathrm{E}\mathrm{K}}$ with $\mathrm{E}\left(3,10\right),\mathrm{K}\left(4,4\right)$

  $\begin{array}{l}\mathrm{E}\mathrm{K}=\sqrt{{\left({\mathrm{x}}_2-{\mathrm{x}}_1\right)}^2+{\left({\mathrm{y}}_2-{\mathrm{y}}_1\right)}^2}\\ \mathrm{E}\mathrm{K}=\sqrt{{\left(4-3\right)}^2+{\left(4-10\right)}^2}\\ \mathrm{E}\mathrm{K}=\sqrt{1+36}\\ \mathrm{E}\mathrm{K}=\sqrt{37}\end{array}$

Find the length of $\overline{\mathrm{S}\mathrm{K}}$ with $\mathrm{S}\left(-1,3\right),\mathrm{K}\left(4,4\right)$

  $\begin{array}{l}\mathrm{S}\mathrm{K}=\sqrt{{\left({\mathrm{x}}_2-{\mathrm{x}}_1\right)}^2+{\left({\mathrm{y}}_2-{\mathrm{y}}_1\right)}^2}\\ \mathrm{S}\mathrm{K}=\sqrt{{\left(4-\left(-1\right)\right)}^2+{\left(4-3\right)}^2}\\ \mathrm{S}\mathrm{K}=\sqrt{25+1}\\ \mathrm{S}\mathrm{K}=\sqrt{26}\end{array}$

Since the corresponding sides of the two triangles are equal. two triangles are congruent by SSS.

Thus, corresponding sides are congruent one obtains $△\mathrm{T}\mathrm{J}\mathrm{D}\cong △\mathrm{S}\mathrm{E}\mathrm{K}$