Std 8 Mathematics · Chapter 4

4. Altitudes and Medians of a Triangle

Solutions to Practice Set 4.1, with construction steps and figures. Maharashtra State Board (Balbharati), Standard Eight Mathematics.

Practice Set 4.1

1In △LMN, ...... is an altitude and ...... is a median. (Write the names of appropriate segments.)

seg LX ⊥ side MN, and Y is the midpoint of side MN.

In △LMN, seg LX is an altitude and seg LY is a median.

seg LX is drawn from vertex L perpendicular to the opposite side MN, so it is an altitude.
seg LY joins vertex L to Y, the midpoint of the opposite side MN, so it is a median.

2Draw an acute angled △PQR. Draw all of its altitudes. Name the point of concurrence as ‘O’.

Step 1: Draw an acute angled △PQR (all three angles less than 90°).

Step 2: Using a set-square, draw a perpendicular from P to side QR. Name its foot D. seg PD is an altitude.

Step 3: Similarly draw seg QE ⊥ side PR and seg RF ⊥ side PQ.

Step 4: The three altitudes meet in one point. Name it O. O is the orthocentre of △PQR.

The orthocentre O of an acute angled triangle lies inside the triangle.

━ altitude- - - extended line

Observation: for an acute angled triangle, the orthocentre is in the interior of the triangle.

3Draw an obtuse angled △STV. Draw its medians and show the centroid.

Step 1: Draw an obtuse angled △STV. Here ∠S is obtuse (more than 90°).

Step 2: Find the midpoint C of side ST (by measuring, or by drawing the perpendicular bisector of ST). Draw seg VC.

Step 3: Similarly find the midpoint A of side TV and draw seg SA, and the midpoint B of side SV and draw seg TB.

Step 4: The three medians meet in one point. Name it G. G is the centroid of △STV.

Equal tick marks show the two halves of each side. G lies inside the triangle.

━ median

Observation: even in an obtuse angled triangle, the centroid is in the interior of the triangle.

4Draw an obtuse angled △LMN. Draw its altitudes and denote the orthocentre by ‘O’.

Step 1: Draw an obtuse angled △LMN. Here ∠L is obtuse.

Step 2: From L, draw seg LP ⊥ side MN. This altitude falls inside the triangle.

Step 3: The perpendiculars from M and N fall outside the triangle. So extend side NL beyond L and side ML beyond L. Draw seg MQ ⊥ line NL and seg NR ⊥ line ML.

Step 4: The three altitudes do not meet each other. Extend the lines containing them (dotted). These three lines meet in one point outside the triangle. Name it O. O is the orthocentre of △LMN.

The lines containing the altitudes meet at O, outside the triangle, beyond the obtuse angle.

━ altitude- - - extended line

Observation: for an obtuse angled triangle, the orthocentre is in the exterior of the triangle, on the side of the obtuse angle.

5Draw a right angled △XYZ. Draw its medians and show their point of concurrence by G.

Step 1: Draw a right angled △XYZ with ∠Y = 90°.

Step 2: Find the midpoint P of side YZ and draw seg XP.

Step 3: Find the midpoint Q of side XZ and draw seg YQ.

Step 4: Find the midpoint R of side XY and draw seg ZR.

Step 5: The three medians meet in one point. Name it G. G is the centroid of △XYZ.

G, the point of concurrence of the medians, lies inside the right angled triangle.

━ median

In a right angled triangle, the median to the hypotenuse (seg YQ) is half as long as the hypotenuse, because Q is at equal distance from X, Y and Z.

6Draw an isosceles triangle. Draw all of its medians and altitudes. Write your observation about their points of concurrence.

Step 1: Draw an isosceles △ABC with AB = AC.

Step 2: Draw the three medians and mark their point of concurrence G (the centroid).

Step 3: Draw the three altitudes and mark their point of concurrence O (the orthocentre).

The median and the altitude from A are the same segment AD. G and O both lie on it.

━ median━ altitude

Observations:

The median from A (the vertex where the equal sides meet) and the altitude from A are the same segment, seg AD. D is the midpoint of BC and also the foot of the perpendicular from A.
The medians from B and C are different from the altitudes from B and C.
So the centroid G and the orthocentre O are different points, but both lie on seg AD, the line of symmetry of the triangle.

If the triangle is equilateral, all three medians coincide with the altitudes, so G and O become the same point.

7Fill in the blanks. Point G is the centroid of △ABC.

AP, BQ and CR are medians meeting at the centroid G.

The centroid divides each median in the ratio 2 : 1. The longer part is from the vertex to G.

(1) If l(RG) = 2.5 then l(GC) = ......

GC : RG = 2 : 1 ∴ l(GC) = 2 × 2.5 = 5

Answer: l(GC) = 5

(2) If l(BG) = 6 then l(BQ) = ......

BG : GQ = 2 : 1 ∴ l(GQ) = 6 ÷ 2 = 3

l(BQ) = l(BG) + l(GQ) = 6 + 3 = 9

Answer: l(BQ) = 9

(3) If l(AP) = 6 then l(AG) = ..... and l(GP) = .....

AP is divided in the ratio 2 : 1, that is, into 3 equal parts. One part = 6 ÷ 3 = 2.

l(AG) = 2 × 2 = 4 and l(GP) = 1 × 2 = 2

Answer: l(AG) = 4, l(GP) = 2

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