In a triangle, two of the sides have lengths 13 and 20. The third side has a length of s, … — ACT

In a triangle, two of the sides have lengths $13$ and $20$ . The third side has a length of $s$ , where $s$ is an integer. If the side of length $20$ is the longest side of the triangle, and the triangle is obtuse, what is the number of possible values for $s$ ?

Cevap: 8

Cevap

There are 8 possible integer values for

s

To find the number of possible integer values for

s

, we combine the Triangle Inequality Theorem (

13 + s > 20 \implies s > 7

) and the condition for an obtuse triangle with

20

as the longest side (

20^2 > 13^2 + s^2 \implies s^2 < 231 \implies s \leq 15

). This limits

s

to integers in the range

[8, 15]

, which contains exactly

8

values.

Adım Adım Çözüm

Apply the Triangle Inequality Theorem to find the lower bound for

s

s > 7

, so the minimum integer value is

8

The sum of the two shorter sides of a triangle must be strictly greater than the longest side.

Set up the obtuse triangle inequality with

20

as the longest side.

20^2 > 13^2 + s^2

In any obtuse triangle with longest side

c

, the inequality

c^2 > a^2 + b^2

must hold.

Solve the inequality

20^2 > 13^2 + s^2

for

s

s^2 < 231 \implies s \leq 15

Simplifying the inequality gives

400 > 169 + s^2 \implies s^2 < 231

. The largest integer whose square is less than

231

15

Determine the number of integers in the range

[8, 15]

8 possible values

The integers satisfying both conditions are

\{8, 9, 10, 11, 12, 13, 14, 15\}

, which count to

8

Anahtar Kavram

Triangle Inequality Theorem and obtuse triangle classification using side lengths

Alternatif Yöntem

List the perfect squares and verify which ones satisfy both

s^2 < 231

and the Triangle Inequality Theorem

s > 7

Tahmini Süre:2m 0s

Soru

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Alternatif Yöntem

Konular

Bu konudan daha fazla soru