Question

!50 POINTS! (2 SIMPLE GEOMETRY QUESTIONS)

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Answer 1

1. Given

2. Reflexive property of congruence

3. HL

x = 20

Explanation:

`\begin{array}l\cline{1-2}\vphantom{\frac12}\sf Statements & \sf Reasons\\\cline{1-2}\vphantom{\frac12} 1.\;\overline{SB} \cong \overline{XG}; \; \triangle BSG\; \textsf{and} \; \triangle XGS\; \sf are\;right\;triangles&1.\;\sf Given\\\cline{1-2}\vphantom{\frac12} 2.\; \overline{SG} \cong \overline{SG} & 2. \; \sf Reflexive\;property\;of\;congruence\\\cline{1-2}\vphantom{\frac12} 3. \; \triangle BSG \cong \triangle XGS & 3.\; \sf HL\\\cline{1-2}\end{array}`

The Reflexive Property of Congruence states that any line segment is congruent to itself. Therefore, as both triangles have the same hypotenuse, line segment SG is congruent to line segment SG.

The Hypotenuse-Leg Triangle Congruence Theorem (HL), states that if the hypotenuse and one leg of a right triangle are congruent to the corresponding parts of another right triangle, then the two triangles are congruent.

`\hrulefill`

We are told that ΔCAP is similar to ΔHIT.

In similar triangles, the corresponding sides are in the same ratio. Therefore:

`\sf \overline{CA} : \overline{HI} = \overline{AP} : \overline{IT} = \overline{CP} : \overline{HT}`

From observation of the given triangles, the side lengths are:

• 
  `\sf \overline{CA} = 4`
• 
  `\sf \overline{AP} = 8`
• 
  `\overline{\sf HI}= a = \sf 10`
• 
  `\overline{\sf IT} = x`

Substituting these into the ratio, we get:

`4 : 10 = 8 : x`

Solve the ratio equation:

`(4)/(10)=(8)/(x)`

`(4)/(10)\cdot 10x=(8)/(x)\cdot 10x`

`4x=80`

`(4x)/(4)=(80)/(4)`

`x=20`

Therefore, the value of x is x = 20.