Chapter 12.5, Problem 22E

To determine

To prove: $△HFS\cong △FST\cong △STH$

Explanation of Solution

Given information:

  

The distances between consecutive bases on a softball field are the same. The distance from home plate to second base is the same as the distance from first base to third base. The angles created at each base are ${90}^{\circ }.$

Formula used:

The Hypotenuse-Leg Congruence Theorem

If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and a leg of a second right triangle , then the two triangles are congruent.

Calculation:

According to the question,

We have,

1. The distances between consecutive bases on a softball field are the same.

  $HF\cong FS\cong ST\cong TH.$

2. The distance from home plate to second base is the same as the distance from first base to third base.

Joining home plate (H) and second base (S), we get two triangles $△HFS\text{ and }△STH\text{.}$

Joining first base (F) and second base (S), we get two triangles $△FST\text{ and }△THF\text{.}$

3. The angles created at each base are ${90}^{\circ }.$

  $\angle H=\angle F=\angle S=\angle T={90}^{\circ }$

Now, proving the triangles are congruent:-

First, consider two triangles:- $△HFS\text{ and }△FST.$

  

Statements	Reasons
H 1. $\overline{HS}=\overline{FT}$ 2. $\angle F\text{ and }\angle S\text{ are right angles}\text{.}$ 3. $△HFS\text{ and }△FST\text{ are right triangles}\text{.}$	1.Given 2. Given. 3. Definition of a right triangle.
L 4. $\overline{FS}=\overline{FS}$ 5. $△HFS\cong △FST$	4. Reflexive Property of Congruence 5. HL Congruence Theorem

Second, consider two triangles:- $△FST\text{ and }△STH.$

  

Statements	Reasons
H 1. $\overline{SH}=\overline{FT}$ 2. $\angle T\text{ and }\angle S\text{ are right angles}\text{.}$ 3. $△FST\text{ and }△STH\text{ are right triangles}\text{.}$	1.Given 2. Given. 3. Definition of a right triangle.
L 4. $\overline{ST}=\overline{ST}$ 5. $△FST\cong △STH$	4. Reflexive Property of Congruence 5. HL Congruence Theorem

Third, consider two triangles:- $△HFS\text{ and }△STH$

  

Statements	Reasons
H 1. $\overline{SH}=\overline{HS}$ 2. $\angle T\text{ and }\angle F\text{ are right angles}\text{.}$ 3. $△HFS\text{ and }△STH\text{ are right triangles}\text{.}$	1.Given 2. Given. 3. Definition of a right triangle.
L 4. $\overline{HS}=\overline{SH}$ 5. $△HFS\cong △STH$	4. Reflexive Property of Congruence 5. HL Congruence Theorem

Therefore, the three triangles are congruent.

  $△HFS\cong △FST\cong △STH\text{.}$

Hence Proved.