3 Marks Question

Question 13 Marks

In the we have AC = DC, CB = CE. Show that AB = DE.

Answer

Given, AC = DC …(i)
and CB = CE …(ii)
According to Euclid’s axiom, if equals are added to equals, then wholes are also equal. So, on adding Eqs.(i) and (ii), we get
AC + CB = DC + CE
⇒ AB = DE

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Question 23 Marks

Read the following axioms:

Things which are equal to the same thing are equal to one another.
If equals are added to equals, the wholes are equal.
Things which are double of the same thing are equal to one another.

Check whether the given system of axioms is consistent or inconsistent.

Answer

Some of Euclid’s axioms are:

Things which are equal to the same thing are equal to one another.
If equals are added to equals, the wholes are equal.
Things which are double of the same things are equal to one another.

Thus, given three axioms are Euclid’s axioms. So, here we cannot deduce any statement from these axioms which contradicts any axiom. So, given system of axioms is consistent.

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Question 33 Marks

In the we have X and Y are the mid-points of AC and BC and AX = CY. Show that AC = BC.

Answer

Given, X is the mid-point of AC
$\text{AX}=\text{CX}=\frac{1}{2}\text{AC}$
$\Rightarrow\ 2\text{AX}=2\text{CX}=\text{AC}\ ...(\text{i})$
And Y is the mid-point of BC
$\text{BY}=\text{CY}=\frac{1}{2}\text{BC}$
$\Rightarrow\ 2\text{BY}=2\text{CY}=\text{BC}\ ...(\text{ii})$
Also, given AX = CY .....(iii)
According to Euclid’s axiom, things which are double of the same things are equal to one another.
From Eq. (iii), 2AX = 2CY
⇒ AC = BC [from Eqs. (i) and (ii)]

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Question 43 Marks

Solve the following question using appropriate Euclid’s axiom:
Two salesmen make equal sales during the month of August. In September, each salesman doubles his sale of the month of August. Compare their sales in September.

Answer

Let the sales of two salesmen in the month of August be x and y. As, they make equal sale during the month of August, x = y. In September, each salesman double his sale of the month of August, So 2x = 2y.
Now, by Euclid’s axiom, thing which are double of the same things are equal to one another.
Hence, we can say that in the month of September also, two salesmen make equal sales.

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Question 53 Marks

In the we have $\text{BX}=\frac{1}{2}\text{AB},\text{ BY}=\frac{1}{2}\text{BC}$ and AB = BC. Show that BX = BY.

Answer

Given, $\text{BX}=\frac{1}{2}\text{AB}$
$\Rightarrow2\text{BX}=\text{AB}\ ...(\text{i})$
$\Rightarrow\text{BY}=\frac{1}{2}\text{BC}$
$\Rightarrow2\text{BY}=\text{BC}\ ...(\text{ii})$
and $\text{AB}=\text{BC}\ ...(\text{iii})$
On putting the values from Eqs. (i) and (ii) in Eq. (iii), we get
2BX = 2BY
According to Euclid’s axiom, things which are double of the same things are equal to one another.
BX = BY

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