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Let’s answer it.

See solution on the comment section.

MTAP (2015 Grade 9)

In Figure 3 below, ABCD is an isosceles trapezoid with AB as its longest side and O divides the diagonals AC and BD in the ratio 1 : 2. What is the area of ABCD if the area of BOC is 2 m^2?

Solution:

Let x be the lengths of DO and CO
2x be the lengths of AO and BO

Since the trapezoid is isosceles, areas of triangles BOC and AOD must be equal.

Area of AOD = 2m^2

The altitude of triangle COD is the same as triangle COB, since the base of COB is 2x which twice the base of COD (x) therefore the area of COD must be half of COB.

Area of COD = (1/2)(2m^2) = 1m^2

The altitude of triangle AOD is the same as triangle AOB, since the base of AOB is 2x which twice the base of AOD (x) therefore the area of AOB must be twice of the area of AOD.

Area of AOB = 2(2m^2) = 4m^2

Area of ABCD = Sum of areas of the four triangles
= 2m^2 + 2m^2 + 1m^2 + 4m^2
= 9m^2

Happy to serve po ☺️

What day of the week is January 1, 1901?

Okay! I don’t have a knowledge about any formulas for finding this one but I can use the modulo of 7 because there are only 7 days in a week.

You can always use the day today, which is November 21, 2022. Then we can count backwards on how many years elapsed from 1901 to 2022.

2022 - 1901 = 121

Therefore, there are roughly 121(365) days from November 21, 1901 to November 21, 2022. (Note that the leap days are not included here)

In order for us to get the number of leap years, the first leap year from 1901 is year 1904. While the leap year closest to 2022 is 2020. Using arithmetic sequence, knowing that the difference between leap years is mostly 4:
a_n = a_1 + (n - 1)d
2020 = 1904 + (n - 1)4

Solving for n:
n = 30

Therefore, there are additional 30 days if we account the leap days from the leap years.

From January 1 to November 21, there are:

30 + 28 + 31 + 30 + 31 + 30 + 31 + 31 + 30 + 31 + 21 = 324 days elapsed

therefore, there are 121(365) + 30 + 324 days from January 1, 1901 to November 21, 2022.

121mod7 = 2
365mod7 = 1
30mod7 = 2
324mod7 = 2

(121(365) + 33 + 324)mod7
= (2(1) + 2 + 2)mod7
= 6mod7

count 6 days backwards from Monday

Sunday
Saturday
Friday
Thursday
Wednesday
Tuesday (answer)

Limits with indeterminate form of infinity minus infinity

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Since HEAT is a rectangle, HE = AT = 21

let point B be in between vertices H and E that the triangle BAT intersects side HE.
point C be the point in between vertices T and A that intersects the altitude of triangle BAT from vertex B
h be the height of the rectangle HEAT
x be the length of TC

h = HT = EA

By Pythagorean theorem
h^2 = 10^2 - x^2
h^2 = 17^2 - (21 - x)^2

100 - x^2 = 42x - x^2 - 152
42x = 252
x = 6

h = sqrt(10^2 - 6^2) = 8

area of rectangle = bh
= 21(8)
= 168