# Math Help - polygons and triangles

1. ## polygons and triangles

2. Question 1:
$AC=\sqrt{AB^2+AC^2}=\sqrt{2}$
$AD=\sqrt{AC^2+CD^2}=\sqrt{2+1}=\sqrt{3}$
$AE=\sqrt{AD^2+DE^2}=\sqrt{3+1}=2$
$AF=\sqrt{AE^2+EF^2}=\sqrt{4+1}=\sqrt{5}$

3. Question 2:
$\widehat{A}+\widehat{B}+\widehat{C}=180^{\circ}\Ri ghtarrow 2(x+y+z)=180^{\circ}\Rightarrow x+y+z=90^{\circ}$
$\widehat{ADB}=180^{\circ}-2z=180^{\circ}-2(90^{\circ}-(x+y))=$
$=180^{\circ}-180^{\circ}+2(x+y)=2\widehat{ACB}$

4. Question 3:
If $\displaystyle \frac{AC}{AB}=k$ then $\displaystyle \overrightarrow{OM}=\frac{1}{1+k}\cdot\overrightar row{OC}+\frac{k}{1+k}\cdot\overrightarrow{OB}$
In this case $\displaystyle k=\frac{7}{3}$

5. Question 4:
$\widehat{FBG}$ is exterior to triangle $BDG$. So $\widehat{FBG}>\widehat{BDG}$ (1).
$\widehat{BDG}$ is exterior to triangle $ADE$. So $\widehat{BDG}>\widehat{AED}$ (2).
From (1) and (2) via tranzitivity, yields $\widehat{FBC}>\widehat{AED}$

6. Hello,

to question 2:

I've modified your drawing (see attachment):

The angle in red is an exterior angle of the isosceles triangle ACD;
the angle in blue is an exterior angle of the isosceles triangle ABD.

The angle at the centre is 2x + 2y = 2(x+y) that means it is twice as large as the angle at C.

7. Hello, Raiden_11!

Basically the same as red_dog . . .

In $\Delta ABC\!:\;2x + 2y + 2z \;=\;180^o\quad\Rightarrow\quad x + y + z \;=\;90^o$

. . $\angle BCA \;=\;x + y \;=\;90^o - z$

In $\Delta ADB\!:\;\angle ADB + 2z \;=\;180^o$

Therefore: . $\angle ADB \;=\;180 - 2z \;=\;2(90^o-z) \;=\;2\cdot\angle BCA$

8. ## Can u help me with these?

2. Observe the relationship between the elements of Diagonal 1 and Diagonal 2 in Pascal’s Triangle, shown on the next page. The sum of the first two elements in Diagonal 1 is equal to the second element in Diagonal 2. Similarly, the sum of the first 5 elements of Diagonal 1 is equal to the 5th element of Diagonal 2. Suppose we wish to use Pascal’s Triangle to find the sum of the first n natural numbers. Use combinatorial notation to express the sum of the n elements of Diagonal 1 in terms of the corresponding value in Diagonal 2.

3. Using the method of mathematical induction, prove that the following statements are false.

9. 2.
$C_1^1+C_2^1=C_3^2$
$C_1^1+C_2^1+C_3^1+C_4^1+C_5^1=C_6^2$
$C_1^1+C_2^1+\ldots C_n^1=C_{n+1}^2$

3.
For the first identity:
For $n=1\Rightarrow 2=2$
For $n=2\Rightarrow 6=6$
For $n=3\Rightarrow 12=14$, false. So the identity is not true for all $n\in\mathbf{N^*}$.

For the second identity:
For $n=1\Rightarrow 2=4$, false. So the identity is not true for all $n\in\mathbf{N^*}$.