Question

Please explain to me how this question can be solved, thank you!

Segment AD is an altitude of triangle ABC.If AD = 12, DC = 5, and AC = 13, find BA. Round to the tenths place if necessary.

1) 33.4
2) 31.2
3) 5.4
4) 4.6

Answer 1

2) 31.2

Explanation:

First let's verify
`\sf \triangle ADC \sim \triangle BAC`

we can use the Pythagorean Theorem, as follows:

`\sf AC^2 = AD^2 + DC^2`

Given that
`\sf AC = 13`,
`\sf AD = 12`, and
`\sf DC = 5`, we can substitute these values into the equation:

`\sf 13^2 = 12^2 + 5^2`

`\sf 169 = 144 + 25`

`\sf 169 = 169`

The equation is satisfied, confirming that
`\sf ABC` is a right-angled triangle.

Now, since
`\sf ABC` is a right triangle.

Here

`\sf \angle D \sim \angle A = 90^\circ` Angle

`\sf \angle C \sim \angle C = \textsf{ (common angle)}` Angle

`\sf \angle A \sim \angle B = \textsf{(remaining angle)}` Angle

So

`\sf \triangle ADC \sim \triangle BAC` by AA postulate

Since the corresponding sides of a similar triangle are proportional.

Therefore, we can use the similarity of triangles to set up the proportion:

`\boxed{\boxed{\sf (BA)/(AD) = (AC)/(DC) =(BC)/(AC)}}`

Taking two of them:

`\sf (BA)/(AD) = (AC)/(DC)`

Substitute the given values:

`\sf (BA)/(12) = (13)/(5)`

Now, solve for
`\sf BA`:

`\sf BA = 12 * (13)/(5)`

`\sf BA = 31.2`

So, the length of
`\sf BA` is 31.2 units.

Therefore, the correct answer from the given options is 2) 31.2.

Answer 2

2) 31.2

Explanation:

The altitude of a triangle is a perpendicular line segment drawn from a vertex to its opposite side. If an altitude is drawn from the right angle of a right triangle, it divides the triangle into two smaller triangles that are similar to both the original larger right triangle and each other.

In similar triangles, corresponding sides are always in the same ratio.

For triangle ABC:

• BA is the longer leg.
• AC is the shorter leg.

For triangle DAC:

• DA is the longer leg.
• DC is the shorter leg.

As the triangles are similar, their corresponding sides are always in the same ratio, so:

`\sf Longer\;leg:Shorter\;leg=BA:AC=DA:DC`

Therefore:

`\sf (BA)/(AC)=(DA)/(DC)`

Given DA = 12, DC = 5 and AC = 13 then:

`\sf (BA)/(13)=(12)/(5)`

Therefore, the length of BA is:

`\sf (BA)/(13)\cdot 13=(12)/(5)\cdot 13`

`\sf BA=31.2`

Therefore, the length of segment BA is 31.2 units.