Two sides of a triangle have lengths 10 and 15 what must be true about the length of the third side

Two sides of a triangle have lengths 10 and 15 what must be true about the length of the third side

Two sides of a triangle have lengths 10 and 15 what must be true about the length of the third side
Two sides of a triangle have lengths 10 and 15 what must be true about the length of the third side

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Consider the triangle. A triangle has angle measures of 45 degrees, 45 degrees, and 90 degrees. Which statement is true about the lengths of the sides? Each side has a different length. Two sides have the same length, which is less than the length of the third side. The three sides have the same length. The sum of the lengths of two sides is equal to the length of the third side.

Two sides of a triangle have lengths

Two sides of a triangle have lengths 10 and 15 what must be true about the length of the third side
and
Two sides of a triangle have lengths 10 and 15 what must be true about the length of the third side
. The length of the altitude to the third side is the average of the lengths of the altitudes to the two given sides. How long is the third side?


Two sides of a triangle have lengths 10 and 15 what must be true about the length of the third side

Solution 1 (Process of Elimination)

The shortest side length has the longest altitude perpendicular to it. The average of the two altitudes given will be between the lengths of the two altitudes, therefore the length of the side perpendicular to that altitude will be between

Two sides of a triangle have lengths 10 and 15 what must be true about the length of the third side
and
Two sides of a triangle have lengths 10 and 15 what must be true about the length of the third side
. The only answer choice that meets this requirement is
Two sides of a triangle have lengths 10 and 15 what must be true about the length of the third side
.

Solution 2

Let the height to the side of length

Two sides of a triangle have lengths 10 and 15 what must be true about the length of the third side
be
Two sides of a triangle have lengths 10 and 15 what must be true about the length of the third side
, the height to the side of length 10 be
Two sides of a triangle have lengths 10 and 15 what must be true about the length of the third side
, the area be
Two sides of a triangle have lengths 10 and 15 what must be true about the length of the third side
, and the height to the unknown side be
Two sides of a triangle have lengths 10 and 15 what must be true about the length of the third side
.

Because the area of a triangle is

Two sides of a triangle have lengths 10 and 15 what must be true about the length of the third side
, we get that
Two sides of a triangle have lengths 10 and 15 what must be true about the length of the third side
and
Two sides of a triangle have lengths 10 and 15 what must be true about the length of the third side
, so, setting them equal,
Two sides of a triangle have lengths 10 and 15 what must be true about the length of the third side
. From the problem, we know that
Two sides of a triangle have lengths 10 and 15 what must be true about the length of the third side
. Substituting, we get that
Two sides of a triangle have lengths 10 and 15 what must be true about the length of the third side
Thus, the side length is going to be
Two sides of a triangle have lengths 10 and 15 what must be true about the length of the third side
.

Solution 3 (Answer Choices)

Let be the height of triangle when the base is

Two sides of a triangle have lengths 10 and 15 what must be true about the length of the third side
and is the height of the triangles when the base is
Two sides of a triangle have lengths 10 and 15 what must be true about the length of the third side
. This means the height for when the triangles has the rd side length, the height would be
Two sides of a triangle have lengths 10 and 15 what must be true about the length of the third side
giving us the following equation:

Two sides of a triangle have lengths 10 and 15 what must be true about the length of the third side
For , we can plug in the answer choices and check when the ratio of
Two sides of a triangle have lengths 10 and 15 what must be true about the length of the third side
for
Two sides of a triangle have lengths 10 and 15 what must be true about the length of the third side
and
Two sides of a triangle have lengths 10 and 15 what must be true about the length of the third side
is the same because we can observe that only for one value, the ratio remains constant. From brute force we get that
Two sides of a triangle have lengths 10 and 15 what must be true about the length of the third side
meaning that
Two sides of a triangle have lengths 10 and 15 what must be true about the length of the third side
is the rd base. ~ Batmanstark

Video Solution

https://youtu.be/27TpizTnMeM

~savannahsolver

See Also

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.

Two sides of a triangle have lengths 10 and 15 what must be true about the length of the third side

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