In quadrilateral ABCD, diagonal AC divides the figure into two triangles, \triangle ABC an… — ACT

In quadrilateral $ABCD$ , diagonal $AC$ divides the figure into two triangles, $\triangle ABC$ and $\triangle ACD$ . The lengths of three of the sides are $AB = 6\text{ cm}$ , $BC = 8\text{ cm}$ , and $CD = 12\text{ cm}$ . If the length of side $AD$ is an integer $k\text{ cm}$ , what is the sum of the minimum and maximum possible values of $k$ ?

A
20
B
24
C
25
26Cevap
E
28

Cevap

To find the minimum and maximum values of the integer side

k

, we first determine the range of the shared diagonal

AC

. In the bottom triangle, the Triangle Inequality Theorem requires

2 < AC < 14

. In the top triangle, the theorem requires

|12 - AC| < k < 12 + AC

. To minimize

k

, we look at the lower bound

|12 - AC|

. Since

AC

can be

12

, the lower bound can be

0

, meaning

k > 0

, so the minimum integer value is

1

. To maximize

k

, we look at the upper bound

12 + AC

. Since

AC < 14

, the upper bound is

k < 26

, so the maximum integer value is

25

. The sum of these values is

1 + 25 = 26

Adım Adım Çözüm

Find the possible range of lengths for the diagonal

AC

using

\triangle ABC

2\text{ cm} < AC < 14\text{ cm}

According to the Triangle Inequality Theorem, the length of side

AC

must be greater than

|BC - AB| = 8 - 6 = 2\text{ cm}

and less than

BC + AB = 8 + 6 = 14\text{ cm}

Set up the inequality for side

AD = k

\triangle ACD

|12 - AC| < k < 12 + AC

By the Triangle Inequality Theorem applied to

\triangle ACD

, the length

k

must be greater than the absolute difference

|CD - AC|

and less than the sum

CD + AC

Find the minimum possible integer value for

k

k = 1

Since

AC

can be any real number between

2

and

14

, we can choose

AC = 12

, which makes the lower bound

|12 - 12| = 0

. Thus, we have

k > 0

. The smallest integer greater than

0

1

. We verify that

k = 1

is possible by choosing

AC = 12

, which satisfies the inequalities for both triangles.

Find the maximum possible integer value for

k

k = 25

Since

AC < 14

, the upper bound is

k < 12 + AC < 12 + 14 = 26

. Thus, we have

k < 26

. The largest integer less than

26

25

. We verify that

k = 25

is possible by choosing

AC = 13.5

, which satisfies the inequalities for both triangles.

Calculate the sum of the minimum and maximum possible integer values of

k

1 + 25 = 26

To find the sum of the minimum and maximum values of

k

as requested by the question.

Anahtar Kavram

The Triangle Inequality Theorem states that for any triangle, the sum of the lengths of any two sides must be strictly greater than the length of the remaining side.

Soru

Cevap

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Anahtar Kavram

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