Your teacher will grade your response to ensure that you receive proper credit for your answer.%0D%0AIt is the same distance from second base to first base, and from second base to third base. The angle formed by first base, second base, and home plate has the same measure as the angle formed by third base, second base, and home plate. What can you conclude about the distance from first base to home plate, and from home plate to third base? Explain using your knowledge of congruent triangles.
In baseball, the bases form a diamond shape that ideally consists of right angles and equal sides. For the purpose of this explanation, we will assume that the shape is a perfect square or at least a rectangle, which is the standard for the infield of a baseball diamond.
According to the given information:
- It's the same distance from second base to first base as it is from second base to third base.
- The angle formed by first base, second base, and home plate is the same measure as the angle formed by third base, second base, and home plate.
This information implies that two triangles are formed: one triangle with vertices at first base, second base, and home plate (let’s call it triangle FSH), and another triangle with vertices at third base, second base, and home plate (let’s call it triangle TSH).
Given the symmetry in the distances and angles:
- Side FS (first to second) is congruent to side TS (third to second) because it's the same distance from second base to first base as it is from second base to third base.
- The angle at second base, angle FST, is shared by both triangles FSH and TSH, and is thus congruent.
- The angle at first base, angle SFH, is congruent to the angle at third base, angle STH, by the given information.
Since the triangles have two congruent angles, and by the properties of triangles, the sum of angles in each triangle is 180 degrees, the third angle at home plate for both triangles must also be congruent (angle SHF congruent to angle SHT).
This tells us that the three corresponding angles in the two triangles are congruent. When two triangles have all corresponding angles congruent, they are similar triangles. And since the distance between second base to the other bases is equal, it means that the scale factor for similarity is 1, which means the triangles are not only similar but also congruent.
In congruent triangles, all corresponding sides are congruent. Thus, since triangle FSH is congruent to triangle TSH, their corresponding sides are congruent:
- SH (home to second) is congruent to itself by the reflexive property.
- FS (first to second) is congruent to TS (third to second), as given.
- This means that FH (first to home) must be congruent to TH (third to home).
So, the distance from first base to home plate is the same as the distance from home plate to third base. This is consistent with the design of a baseball diamond, where all the bases are 90 feet apart, forming a square (assuming a regulation infield).