Question

In this figure, DC=5, and BC=3.

What is the length of the longer leg of △ADB?

Enter your answer rounded to the nearest tenth in the box.

length of longer leg =

Answer 1

The length of AD is
`\(√(40)\)`, rounded to the nearest tenth. Therefore, the length of the longer leg is approximately 6.3.

In triangle ADB, where AC is the angle bisector to DB, and angles ACB, DAB, and ACD are each 90 degrees, we can apply the angle bisector theorem and the Pythagorean theorem to find the length of AD.

Let x be the length of AD. According to the angle bisector theorem:

`\[(AC)/(BC) = (AD)/(DB)\]`

Substitute the given values:

`\[(AC)/(3) = (x)/(x + 5)\]`

Cross-multiply:

`\[AC \cdot (x + 5) = 3x\]`

Since ACB is a right angle, we can use the Pythagorean theorem for triangle ABC:

`\[AC^2 + BC^2 = AB^2\]\\\\AC^2 + 3^2 = (x + 5)^2\]`

Solve these equations simultaneously.

After calculations, the length of AD is
`\(√(40)\)`, rounded to the nearest tenth.

Therefore, the length of the longer leg is approximately 6.3.

The probable question may be:

In triangle ADB , AC is angle bisector to DB. DC=5, and BC=3. angle ACB=Angle DAB= angle ACD=90 degree

What is the length of the AD of △ADB?

Enter your answer rounded to the nearest tenth in the box.

length of longer leg = ?

Answer 2

The length of the longer leg of triangle ADB is approximately 5.8.

Here's the approach:

Identify the right triangles: AC is perpendicular to DB, creating two right triangles: ACB and ACD.

Apply the Pythagorean theorem:

In a right triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides.

Focus on triangle ACB:

Hypotenuse: AB (the longer leg of triangle ADB)

One leg: BC = 3

Other leg: AC (unknown)

Find AC using triangle ACD:

Hypotenuse: AC (which is also a leg of triangle ACB)

One leg: DC = 5

Other leg: AD (unknown, but not needed for this step)

Apply the Pythagorean theorem: AC² = AD² + DC² = 5² = 25

Solve for AC: AC = √25 = 5

Apply the Pythagorean theorem to triangle ACB:

AB² = AC² + BC² = 5² + 3² = 34

Solve for AB:

AB = √34 ≈ 5.8 (rounded to the nearest tenth)

Question:

In this figure, DC=5, and BC=3.

What is the length of the longer leg of △ADB?

Enter your answer rounded to the nearest tenth in the box.

length of longer leg =