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# In ∆ ABC shown above, AB = BC and base AC is equal to the altitude of

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Math Expert
Joined: 02 Sep 2009
Posts: 52971
In ∆ ABC shown above, AB = BC and base AC is equal to the altitude of  [#permalink]

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15 Dec 2017, 01:05
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Difficulty:

25% (medium)

Question Stats:

78% (02:16) correct 22% (02:22) wrong based on 66 sessions

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In ∆ ABC shown above, AB = BC and base AC is equal to the altitude of the triangle from point B. If the coordinates of points A and C are (2, 5) and (6, 5) respectively, which of the following could be the coordinates of point B?

(A) (2, 7)
(B) (2, 8)
(C) (2, 9)
(D) (4, 8)
(E) (4, 9)

Attachment:

2017-12-15_1258_001.png [ 4.47 KiB | Viewed 815 times ]

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In ∆ ABC shown above, AB = BC and base AC is equal to the altitude of  [#permalink]

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15 Dec 2017, 07:23
2
Hi
My explanation is as per attached sketch.
Please see how we can easily eliminate the choices A, B & C in a few seconds.

if you like the approach, appreciate by kudos.

Bunuel wrote:

In ∆ ABC shown above, AB = BC and base AC is equal to the altitude of the triangle from point B. If the coordinates of points A and C are (2, 5) and (6, 5) respectively, which of the following could be the coordinates of point B?

(A) (2, 7)
(B) (2, 8)
(C) (2, 9)
(D) (4, 8)
(E) (4, 9)

Attachment:
The attachment 2017-12-15_1258_001.png is no longer available

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WhatsApp Image 2017-12-15 at 20.54.16.jpeg [ 102.63 KiB | Viewed 649 times ]

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In ∆ ABC shown above, AB = BC and base AC is equal to the altitude of  [#permalink]

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15 Dec 2017, 09:03
Bunuel wrote:

In ∆ ABC shown above, AB = BC and base AC is equal to the altitude of the triangle from point B. If the coordinates of points A and C are (2, 5) and (6, 5) respectively, which of the following could be the coordinates of point B?

(A) (2, 7)
(B) (2, 8)
(C) (2, 9)
(D) (4, 8)
(E) (4, 9)

Attachment:
2017-12-15_1258_001.png

Four answers can be rejected quickly: A, B, C, and D.

The prompt makes it easy to find B's y-coordinate. We are given that the altitude from point B = base AC. Add the length of AC to 5 to find B's y-coordinate.

Length of AC: (6 - 2) = 4
Points A and C (and AC) are at y=5. Add the 4
(5 + 4) = 9 = B's y-coordinate

Eliminate answers A, B, and D, whose y-coordinates are not 9

Also eliminate Answer C (2, 9), for a number of reasons.* If B is at (2,9), directly above A, we now have a right isosceles triangle where AB = AC and hypotenuse BC does NOT equal AB. BC must = AC. Reject C.

AB = BC: ∆ ABC is isosceles. Its altitude is a perpendicular bisector of the base AC.

The altitude from B hits the base at AC's midpoint (4,5). The x-coordinate for AC's midpoint is (2+6)/2 = 4. The x-coordinate of B must be 4.

B is at (4,9)

*If that doesn't stand out, think of side lengths and angle measures. AB = BC, so
∠A = ∠C (angles opposite equal sides are equal).

Imagine B "moving" to directly above A: ∠A gets bigger, ∠C gets smaller. Not possible; they're equal. Side AB gets shorter, side BC gets longer. Also not possible: AB = BC

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In ∆ ABC shown above, AB = BC and base AC is equal to the altitude of   [#permalink] 15 Dec 2017, 09:03
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