Question

If PR bisects ∠SRT and U is the midpoint of RT.

Answer 1

Answer:

Part 1) Triangle UQR Is an obtuse isosceles triangle

Part 2) Triangle RST is a equilateral triangle

Part 3) Triangle SRQ is a right scalene triangle

Part 4) Triangle PRT Is an obtuse isosceles triangle

Part 5) Triangle TQU is a equilateral triangle

Part 6) Triangle PQT is a right scalene triangle

Step-by-step explanation:

The complete question is:

If PR bisects ∠SRT and U is the midpoint of RT, classify each triangle by its angles and sides

Triangle UQR

Triangle RST

Triangle SRQ

Triangle PRT

Triangle TQU

Triangle PQT

The picture of the question in the attached figure

Part 1) Triangle UQR

we know that

PR bisects ∠SRT

That means

$m\angle SRQ=m\angle QRU$

we have

$m\angle SRQ=30^o$ ---> is given

so

$m\angle QRU=30^o$

Remember that the sum of the interior angles in any triangle must be equal to 180 degrees

so

In the triangle UQR

$m\angle UQR+120^o+30^o=180^o$

$m\angle UQR=180^o-150^o=30^o$

so

Triangle UQR is a  $30^o-120^o-30^o$

Remember that

An isosceles triangle has two equal sides and two equal interior angles

so

$QU=RU=8\ m$

Classify

By its angles is an obtuse triangle (has an interior angle greater than 90 degrees)

By its sides is an isosceles triangle (has two equal sides)

therefore

Triangle UQR Is an obtuse isosceles triangle

Part 2) Triangle RST

we know that

$RU=TU=8\ m$ ---> because U is the midpoint of RT

$m\angle SRT=2m\angle SRQ$ ----> because PR bisects ∠SRT

so

$m\angle SRT=2(30^o)=60^o$

Classify

By its angles is an acute triangle ( interior angles less than 90 degrees)

By its sides is an equilateral triangle (has three equal sides)

therefore

Triangle RST is a equilateral triangle

Note: An equilateral triangle is subtended to be an acute triangle, because the measure of the interior angles is always 60 degrees

Part 3) Triangle SRQ

we know that

Triangle SRQ is a $60^o-90^o-30^o$

Classify

By its angles is a right triangle (has an interior angle equal to 90 degrees)

By its sides is a scalene triangle (has three different sides)

Note: If any triangle has three different interior angles, then the triangle has three different length sides

therefore

Triangle SRQ is a right scalene triangle

Part 4) Triangle PRT

Find the length side QR

Applying the Pythagorean Theorem in the right triangle SRQ

$QR^2=16^2-8^2\\QR^2=192\\QR=\sqrt{192}=13.9\ m$

Find the length side PQ

Applying the Pythagorean Theorem in the right triangle PQT

$PQ^2=16^2-8^2\\PQ^2=192\\PQ=\sqrt{192}=13.9\ m$

The length sides of triangle PRT are

$PR=13.9(2)=27.8\ m\\TP=TR=16\ m$

The interior angle of triangle PRT are $30^o-120^o-30^o$

Classify

By its angles is an obtuse triangle (has an interior angle greater than 90 degrees)

By its sides is an isosceles triangle (has two equal sides)

therefore

Triangle PRT Is an obtuse isosceles triangle

Part 5) Triangle TQU

Find the measure of angle TUQ

we know that

$m\angle TUQ+120^o=180^o$ ----> by supplementary angles (form a linear pair)

$m\angle TUQ=180^o-120^o=60^o$

Find the measure of angle TQU

we know that

$m\angle TQU+m\angle UQR=90^o$ ----> by complementary angles

$m\angle UQR=30^o$ ----> see part 1)

$m\angle TQU=90^o-30^o=60^o$

so

The interior angle of triangle TQU are $60^o-60^o-60^o$

Note: If a triangle has three equal interior angles, then the triangle has three equal sides (QT=QU=TU=8 m)

Classify

Triangle TQU is a equilateral triangle

Note: An equilateral triangle is subtended to be an acute triangle, because the measure of the interior angles is always 60 degrees

Part 6) Triangle PQT

we know that

Triangle PQT and Triangle RQS are congruent by SSS postulate

so

The interior angle of triangle PQT are $60^o-90^o-30^o$

The length sides of triangle PQT are $8\ m-16\ m-13.9\ m$

Classify

By its angles is a right triangle (has an interior angle equal to 90 degrees)

By its sides is a scalene triangle (has three different sides)

Note: If any triangle has three different interior angles, then the triangle has three different length sides

therefore

Triangle PQT is a right scalene triangle