Internal angle bisectors of triangle ABC with right angle at B meet at

Question

Internal angle bisectors of triangle ABC with right angle at B meet at a point P inside the triangle. What is the perpendicular distance from point P to side AB?

(1) Length of side AB and BC is 8 cm and 6 cm respectively

(2) The perimeter of the triangle ABC is 24 cm

chetan2u · Accepted Answer

Purely from the perspective of DS If you have a UNIQUE triangle, you would be able to find almost all the related terms, be it angles, area, incenter, etc. Now, we are given a right angled triangle with B as 90. (1) Length of side AB and BC is 8 cm and 6 cm respectively AC is the hypotenuse and 10 in length, because sides are in ratio 6:8:10 or 3:4:5. Fixed and unique triangle. Sufficient (2) The perimeter of the triangle ABC is 24 cm. We can have various triangles as sides need not be integer. Insufficient A PS solution . When you join all internal angle bisectors as given in sketch, we have a point O, which is called incenter. Incenter will give us the radius of the incircle. Untitled02.png The perpendicular distance from point P to side AB is nothing but the radius r. (1) Length of side AB and BC is 8 cm and 6 cm respectively So we have a 6:8:10 triangle. Area of riangle ABC = \frac{1}{2}*8*6=24 ...(i) If we take individual inner triangles, Area of riangle ABC = A( riangle ABO)+A( riangle ACO)+A( riangle CBO)= \frac{1}{2}*8*r+\frac{1}{2}*r*6+\frac{1}{2}*10*r=\frac{1}{2}*24*r=12 ...(ii) From i and ii, 12r=24 or r=2 Sufficient (2) The perimeter of the triangle ABC is 24 cm Various possibilities Insufficient A

KarishmaB · Answer

The triangle ABC has one right angle. We don't know the other angles. The angle bisectors' placements will depend on what the other angles are. Hence, the placement of P will vary. Also, we don't know which leg is AB. We don't know the length of AB and the measure of angle A. So we don't know the distance from P to AB. (1) Length of side AB and BC is 8 cm and 6 cm respectively AB and BC are legs of the triangle since right angle is at B. Their lengths are 6 and 8. This means the hypotenuse is of length 10. Note that this completely defines the triangle. There is only one such triangle we can make. The measure of angles A and C are unique. Hence we can uniquely determine P and its perpendicular distance from AB. This statement alone is sufficient. You need to do nothing else to evaluate this statement. Note that if we have "3 side lengths of any triangle" or "two side lengths with the included angle" or "two angles and the included side", it is uniquely defined. Try constructing a triangle with the given data and see if you can do it uniquely. (2) The perimeter of the triangle ABC is 24 cm You can draw different triangles with perimeter 24 cm. The two sides AB and BC could be equal lengths or AB could be much greater than BC and so on... Screenshot 2021-12-06 at 09.52.16.png As per how we draw the triangle, the placement of P will vary and hence the perpendicular distance from AB will vary. Hence this statement will not give us a unique answer. Answer (A)

shalin23 · Answer

can someone explain this?

shalin23 · Answer

can anyone share the solution for this question?

Karthik740 · Answer

The answers here are amazing and noteworthy.

Namangupta1997 · Answer

Statement 2 gives no new info. Still can't work out the math from the info given in statement 1.

BrentGMATPrepNow  care to weigh in ?

rsrighosh · Answer

Bunuel   chetan2u   VeritasKarishma

Could you please share the solution. Unable to understand how A is the answer.

Fdambro294 · Answer

In many of the very difficult data sufficiency questions involving triangles and other figures, whether the statement is sufficient comes down to whether there is 1 unique figure.

What I mean is, if we know that the triangle ABC is one, specific figure, then there is only one unique value for whatever the question may ask:

perimeter
area
A particular interior angle measure

Or as in this case, the perpendicular distance from the point of intersection of the Angle Bisectors (called the In center, but you don’t have to know this)

We are given that this triangle is a Right Triangle at vertex B.

If we knew the 2 adjacent sides that include Angle <B, the triangle will be “uniquely determined” (S-A-S ——> knowing these 3 unique values fixes the triangle in place)

Statement 1: the legs are 6 and 8 —-> this is a 6-8-10 right triangle.

The triangle is therefore uniquely set in one place. In other words, there is one particular set of interior angles, 3 unique angle Bisectors that meet at point P, etc.

Since we know that the triangle is fixed in place, we could theoretically answer the question with one unique value.

S1 sufficient

S2: all we have is the perimeter. There is no integer constraint for the side lengths.

Therefore, there exists more than one right triangle that has the perimeter of 24.

S2 not sufficient.

A

Note, if you wanted to solve the problem:

The angle Bisectors of the 3 interior angles of a triangle meet at the Incenter.

The Incenter will be the center of the circle that can be inscribed within the right triangle ABC

The perpendicular distance from point P to side AB will be the Inradius for this inscribed circle.

Question is really asking what is the inradius for the circle that can be inscribed perfectly within right triangle with the sides 6-8-10

Step 1: draw the inscribed circle and draw 2 radii from center P to the point of Tangency at which the circle intersects with the Legs of the right triangle (AB and BC)

Call the radii = r

Rule: the radius of a circle drawn to the point of Tangency will be perpendicular to the tangent line.

In this case, r will form a 90 degree angle with both sides AB and BC of the right triangle (the Legs)

Step 2: this creates a Square of side length r around Vertex B

Next rule: from a single exterior point, 2 lines drawn tangent to a circle will be equal.

Therefore:

Side AB = 8

From Vertex A to the point of Tangency on side AB ——> length = 8 - r

8 - r = length of Vertex A to the point of Tangency on the hypotenuse.

Then do the same for Side BC

6 - r = length of Vertex C to the point of Tangency on the hypotenuse

The entire hypotenuse length is:

(8 - r) + (6 - r)

And we know from the fact that this is a Pythagorean Triplet that the hypotenuse us 10

(8 - r) + (6 - r) = 10

14 - 2r = 10

4 = 2r

r = 2

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Internal angle bisectors of triangle ABC with right angle at B meet at

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GMAT Data Sufficiency (DS) Questions

Internal angle bisectors of triangle ABC with right angle at B meet at

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