## Chapter 4 – Altitudes and Medians of a triangle

**Practice set 4.1**

**1. In ∆ LMN, ___ is an altitude and ___ is a median. (write the names of appropriate segments.)**

**Ans:** In ∆ LMN, **___** is an altitude and ___ is a median.

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

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

**Solution:**

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

**Solution:**

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

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

**Solution:**

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

**Solution:**

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

**Solution:**

**The point of concurrence of medians i.e. G and that of altitudes i.e. O lie on the same line PS which is the perpendicular bisector of seg QR.**

**7. Fill in the blanks.**

**7. Fill in the blanks.**

**Point G is the centroid of ∆ ABC.**

**(1) If l(RG) = 2.5 then l(GC) = ___**

**Ans:** The centroid of a triangle divides each median in the ratio 2:1.

**Point G is the centroid and seg CR is the median.**

**∴ \(\large \frac {l(GC)}{l(RG)}\) = \(\large \frac {2}{1}\)**

**∴ \(\large \frac {l(GC)}{2.5}\) = \(\large \frac {2}{1}\)**

**∴ l(GC) × 1 = 2 × 2.5**

**∴ l(GC) = 5**

**(2) If l(BG) = 6 then l(BQ) = ___**

**Ans: **The centroid of a triangle divides each median in the ratio 2:1.

**Point G is the centroid and seg BQ is the median.**

**∴ \(\large \frac {l(BG)}{l(GQ)}\) = \(\large \frac {2}{1}\)**

**∴ \(\large \frac {6}{l(GQ)}\) = \(\large \frac {2}{1}\)**

**∴ 6 × 1 = 2 × l(GQ)**

**∴ \(\large \frac {6}{2}\) = l(GQ)**

**∴ l(GQ) = 3**

**Now, l (BQ) = l(BG) + l(GQ)**

**∴ l(BQ) = 6 + 3**

**∴ l(BQ) = 9**

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

**Ans: **The centroid of a triangle divides each median in the ratio 2:1.

**Point G is the centroid and seg AP is the median.**

**∴ \(\large \frac {l(AG)}{l(GP)}\) = \(\large \frac {2}{1}\)**

**∴ l(AG) = 2 l(GP) …(i)**

**Now, **

**l(AP) = l(AG) + l(GP) …(ii)**

**∴ l(AP) = 2l(GP) + l(GP) … [From (i)]**

**∴ l(AP) = 3l(GP)**

**∴ 6 = 3l(GP) . .[∵ l(AP) = 6]**

**∴ \(\large \frac {6}{3}\) = l(GP)**

**∴ l(GP) = 2**

**And,**

**l(AP) = l(AG) + l(GP) … [from (ii)]**

**∴ 6 = l(AG) + 2**

**∴ l(AG) = 6 – 2**

**∴ l(AG) = 4**